Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      2. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      3. Generalized Absolute Values and Polar Decompositions of a Bounded Operator      Piotr Niemiec 《Integral Equations and Operator Theory》2011,71(2):151-160 Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?K|2 ￡ á|T|x, x?Há|T*|y, y?K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR+ ? \mathbbR+{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.      4. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      5. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      6. From Chaos to Global Convergence      Giovanni Di Lena  Davide Franco  Mario Martelli  Basilio Messano 《Mediterranean Journal of Mathematics》2011,8(4):473-489 The main purpose of this paper is to investigate dynamical systems F : \mathbbR2 ? \mathbbR2{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR2 ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x 0, y 0), such that the orbit O(x0) = {x0, x1 = F(x0), x2 = F(x1), . . . }, O({\rm{x}}_0) = \{{\rm{x}}_0, {\rm{x}}_1 = F({\rm{x}}_0), {\rm{x}}_2 = F({\rm{x}}_1), . . . \},      7. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      8. Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions      Gilbert Helmberg 《Journal of Fourier Analysis and Applications》2002,8(1):29-42 Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR2 : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums Sn,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of Sn,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR2 (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.      9. On the measurable solutions of a functional equation      Antal Járai 《Aequationes Mathematicae》2010,80(1-2):131-139 The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.      10. Linear independence of time-frequency shifts under a generalized Schrödinger representation      G. Kutyniok 《Archiv der Mathematik》2002,78(2):135-144 Let r\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e2 pi y t f(t + x) = (r\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L2(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and rG \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {rG (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L2(G), f 1 0 f \in L^2(G), f \neq 0 .      11. A convexity theorem for multiplicative functions      Pierre Maréchal 《Optimization Letters》2012,6(2):357-362 We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbRm ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β 1 + ··· + β n . We also provide further generalization to functions of the form (x,y1, . . . , yn)? g(x)1+af1(y1)-b1 ···fn(yn)-bn{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f k concave, positively homogeneous and nonnegative on their domains.      12. Ground states for Schrödinger–Poisson type systems      Giusi?VairaEmail author 《Ricerche di matematica》2011,60(2):263-297 In this paper we consider the following elliptic system in \mathbbR3{\mathbb{R}^3} $\qquad\left\{{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\right.$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.      13. New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces      Dachun Yang  Wen Yuan 《Mathematische Zeitschrift》2010,265(2):451-480 Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]s, tp, q(\mathbbRn) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]-s, tp￠, q￠(\mathbbRn){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .      14. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      15. L p Boundedness for the Multilinear Singular Integral Operator      Guoen Hu 《Integral Equations and Operator Theory》2005,52(3):437-449 Lp (\mathbbRn )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by $T^{*}_{A} f(x) = {\mathop {\sup }\limits_{ \in > 0} }{\left| {{\int_{|x - y| > \in } {\frac{{\Omega (x - y)}} {{|x - y|^{{n + 1}} }}} }(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} \right|},$T^{*}_{A} f(x) = {\mathop {\sup }\limits_{ \in > 0} }{\left| {{\int_{|x - y| > \in } {\frac{{\Omega (x - y)}} {{|x - y|^{{n + 1}} }}} }(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} \right|},      16. Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces      K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573 Let W ì \mathbbRn \Omega \subset \mathbb{R}^n be an open set and l(x) | u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}      17. Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}      Ricardo Abreu Blaya  Juan Bory Reyes  Dixan Peña Peña  Frank Sommen 《Advances in Applied Clifford Algebras》2010,20(1):1-12 The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system ?x1 + i [(f)\tilde] ?x 2 = 0\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}      18. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      19. Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$      Boumediene Abdellaoui  Veronica Felli  Ireneo Peral 《Calculus of Variations and Partial Differential Equations》2009,34(1):97-137 We study the existence of different types of positive solutions to problem where , , and is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0. B. Abdellaoui and I. Peral supported by projects MTM2007-65018, MEC and CCG06-UAM/ESP-0340, Spain. V. Felli supported by Italy MIUR, national project Variational Methods and Nonlinear Differential Equations.      20. Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$      A. Arkhipova 《Journal of Mathematical Sciences》2011,176(6):732-758 We prove the existence of a global heat flow u : Ω ×  \mathbbR+ ? \mathbbRN {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR+ {\mathbb{R}^{+}}) ⊂  \mathbbRn {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbRN {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbRN {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Generalized Absolute Values and Polar Decompositions of a Bounded Operator

Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?_K|² ￡ á|T|x, x?_Há|T^*|y, y?_K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR₊ ? \mathbbR₊{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.     

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      5. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      6. From Chaos to Global Convergence      Giovanni Di Lena  Davide Franco  Mario Martelli  Basilio Messano 《Mediterranean Journal of Mathematics》2011,8(4):473-489 The main purpose of this paper is to investigate dynamical systems F : \mathbbR2 ? \mathbbR2{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR2 ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x 0, y 0), such that the orbit O(x0) = {x0, x1 = F(x0), x2 = F(x1), . . . }, O({\rm{x}}_0) = \{{\rm{x}}_0, {\rm{x}}_1 = F({\rm{x}}_0), {\rm{x}}_2 = F({\rm{x}}_1), . . . \},      7. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      8. Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions      Gilbert Helmberg 《Journal of Fourier Analysis and Applications》2002,8(1):29-42 Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR2 : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums Sn,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of Sn,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR2 (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.      9. On the measurable solutions of a functional equation      Antal Járai 《Aequationes Mathematicae》2010,80(1-2):131-139 The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.      10. Linear independence of time-frequency shifts under a generalized Schrödinger representation      G. Kutyniok 《Archiv der Mathematik》2002,78(2):135-144 Let r\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e2 pi y t f(t + x) = (r\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L2(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and rG \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {rG (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L2(G), f 1 0 f \in L^2(G), f \neq 0 .      11. A convexity theorem for multiplicative functions      Pierre Maréchal 《Optimization Letters》2012,6(2):357-362 We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbRm ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β 1 + ··· + β n . We also provide further generalization to functions of the form (x,y1, . . . , yn)? g(x)1+af1(y1)-b1 ···fn(yn)-bn{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f k concave, positively homogeneous and nonnegative on their domains.      12. Ground states for Schrödinger–Poisson type systems      Giusi?VairaEmail author 《Ricerche di matematica》2011,60(2):263-297 In this paper we consider the following elliptic system in \mathbbR3{\mathbb{R}^3} $\qquad\left\{{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\right.$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.      13. New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces      Dachun Yang  Wen Yuan 《Mathematische Zeitschrift》2010,265(2):451-480 Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]s, tp, q(\mathbbRn) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]-s, tp￠, q￠(\mathbbRn){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .      14. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      15. L p Boundedness for the Multilinear Singular Integral Operator      Guoen Hu 《Integral Equations and Operator Theory》2005,52(3):437-449 Lp (\mathbbRn )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by $T^{*}_{A} f(x) = {\mathop {\sup }\limits_{ \in > 0} }{\left| {{\int_{|x - y| > \in } {\frac{{\Omega (x - y)}} {{|x - y|^{{n + 1}} }}} }(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} \right|},$T^{*}_{A} f(x) = {\mathop {\sup }\limits_{ \in > 0} }{\left| {{\int_{|x - y| > \in } {\frac{{\Omega (x - y)}} {{|x - y|^{{n + 1}} }}} }(A(x) - A(y) - \nabla A(y)(x - y))f(y)dy} \right|},      16. Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces      K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573 Let W ì \mathbbRn \Omega \subset \mathbb{R}^n be an open set and l(x) | u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}      17. Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}      Ricardo Abreu Blaya  Juan Bory Reyes  Dixan Peña Peña  Frank Sommen 《Advances in Applied Clifford Algebras》2010,20(1):1-12 The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system ?x1 + i [(f)\tilde] ?x 2 = 0\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}      18. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      19. Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$      Boumediene Abdellaoui  Veronica Felli  Ireneo Peral 《Calculus of Variations and Partial Differential Equations》2009,34(1):97-137 We study the existence of different types of positive solutions to problem where , , and is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0. B. Abdellaoui and I. Peral supported by projects MTM2007-65018, MEC and CCG06-UAM/ESP-0340, Spain. V. Felli supported by Italy MIUR, national project Variational Methods and Nonlinear Differential Equations.      20. Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$      A. Arkhipova 《Journal of Mathematical Sciences》2011,176(6):732-758 We prove the existence of a global heat flow u : Ω ×  \mathbbR+ ? \mathbbRN {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR+ {\mathbb{R}^{+}}) ⊂  \mathbbRn {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbRN {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbRN {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient

We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR² é W? \mathbbR² u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form

Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR² : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums S_n,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of S_n,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR² (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.     

On the measurable solutions of a functional equation

The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.     

Linear independence of time-frequency shifts under a generalized Schrödinger representation

Let r_\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e^{2 pi y t} f(t + x) = (r_\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L²(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and r_G \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {r_G (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L²(G), f 1 0 f \in L^2(G), f \neq 0 .     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Let W ì \mathbbRⁿ \Omega \subset \mathbb{R}^n be an open set and l(x) | u |_p,l = ( ò_W l^p (x)| u(x) |^p dx )^1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}     

Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let X ì \mathbb Rⁿ{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.