Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      2. Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions      Hans-Christoph Grunau  Frédéric Robert 《Archive for Rational Mechanics and Analysis》2010,195(3):865-898 In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRn{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRn{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .      3. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      4. Symmetries of center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2010,13(2):196-227 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR2 O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D+ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D 2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D 2 if and only if the 1-jet of F at O is a “rotation,” i.e., j1F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D+ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.      5. The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity      Craig Cowan  Pierpaolo Esposito  Nassif Ghoussoub  Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787 We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u λ with 0 < u λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.      6. The Instationary Navier–Stokes Equations in Weighted Bessel-Potential Spaces      Katrin Schumacher 《Journal of Mathematical Fluid Mechanics》2009,11(4):552-571 We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain W ì \mathbbRn \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in Lr(0, T; Hb, qw (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where Hb, qw (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.      7. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      8. On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows      Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473 Consider the class of C r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracpnqn| ￡ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions.      9. Stationary Flows of Shear Thickening Fluids in 2D      Martin Fuchs 《Journal of Mathematical Fluid Mechanics》2012,14(1):43-54 We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR2{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).      10. Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation   总被引：4，自引：0，他引：4   Enzo Vitillaro 《Archive for Rational Mechanics and Analysis》1999,149(2):155-182 We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRn\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.      11. Heteroclinic cycles arising in generic unfoldings of nilpotent singularities      Pablo G. Barrientos  Santiago Ibáñez  J. Ángel Rodríguez 《Journal of Dynamics and Differential Equations》2011,23(4):999-1028 In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR4{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR3{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.      12. A New Approach to Equations with Memory      Mauro Fabrizio  Claudio Giorgi  Vittorino Pata 《Archive for Rational Mechanics and Analysis》2010,198(1):189-232 We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR+ ? \mathbbR+{G : \mathbb{R}^+ \to \mathbb{R}^+} such that $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      13. Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-growth      Thomas Schmidt 《Archive for Rational Mechanics and Analysis》2009,193(2):311-337       14. Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise      Viorel Barbu  Michael Röckner 《Archive for Rational Mechanics and Analysis》2013,209(3):797-834 In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.      15. Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary      Jann-Long Chern  Chang-Shou Lin 《Archive for Rational Mechanics and Analysis》2010,197(2):401-432 Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0￥(W){C_0^\infty(\Omega)} with respect to the norm: ||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      16. Symmetries of degenerate center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2009,12(4):522-542 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ì \mathbbR2 O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D+ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D 2 that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D+ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j 1 F(O) of F at O is nondegenerate. In this paper, the degenerate case j 1 F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D+ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D+ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D+ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O.      17. Billiards and Two-Dimensional Problems of Optimal Resistance      Alexander Plakhov 《Archive for Rational Mechanics and Analysis》2009,194(2):349-381 A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR2{B \subset \mathbb{R}^2} we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR2 \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.      18. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      19. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales      S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428 Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation ( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0,    t ? [ t0,￥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}},      20. On Kato’s Method for Navier–Stokes Equations      Bernhard H. Haak  Peer Chr. Kunstmann 《Journal of Mathematical Fluid Mechanics》2009,11(4):492-535 We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR3{\mathbb{R}}^{3} and irregular domains in \mathbbRn{\mathbb{R}}^{n}.      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

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

Symmetries of center singularities of plane vector fields

Let D² ì \mathbbR² {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR² O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D⁺ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D ² that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D ² if and only if the 1-jet of F at O is a “rotation,” i.e., j¹F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D⁺ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

The Instationary Navier–Stokes Equations in Weighted Bessel-Potential Spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain W ì \mathbbRⁿ \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in L^r(0, T; H^{b, q}_w (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where H^{b, q}_w (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.     

Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth

We consider the quasilinear problem

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

Stationary Flows of Shear Thickening Fluids in 2D

We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR²{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).     

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRⁿ\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.     

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR⁴{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR³{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.     

A New Approach to Equations with Memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR⁺ ? \mathbbR⁺{G : \mathbb{R}^+ \to \mathbb{R}^+} such that

$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      13. Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-growth      Thomas Schmidt 《Archive for Rational Mechanics and Analysis》2009,193(2):311-337       14. Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise      Viorel Barbu  Michael Röckner 《Archive for Rational Mechanics and Analysis》2013,209(3):797-834 In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.      15. Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary      Jann-Long Chern  Chang-Shou Lin 《Archive for Rational Mechanics and Analysis》2010,197(2):401-432 Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in RN, N\geqq 3{{\bf R}^N, N\geqq 3}, and Da1,2(W){D_a^{1,2}(\Omega)} be the completion of C0￥(W){C_0^\infty(\Omega)} with respect to the norm: ||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      16. Symmetries of degenerate center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2009,12(4):522-542 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ì \mathbbR2 O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D+ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D 2 that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D+ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j 1 F(O) of F at O is nondegenerate. In this paper, the degenerate case j 1 F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D+ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D+ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D+ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O.      17. Billiards and Two-Dimensional Problems of Optimal Resistance      Alexander Plakhov 《Archive for Rational Mechanics and Analysis》2009,194(2):349-381 A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR2{B \subset \mathbb{R}^2} we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR2 \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.      18. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      19. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales      S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428 Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation ( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0,    t ? [ t0,￥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}},      20. On Kato’s Method for Navier–Stokes Equations      Bernhard H. Haak  Peer Chr. Kunstmann 《Journal of Mathematical Fluid Mechanics》2009,11(4):492-535 We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR3{\mathbb{R}}^{3} and irregular domains in \mathbbRn{\mathbb{R}}^{n}.      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

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      16. Symmetries of degenerate center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2009,12(4):522-542 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ì \mathbbR2 O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D+ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D 2 that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D+ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j 1 F(O) of F at O is nondegenerate. In this paper, the degenerate case j 1 F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D+ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D+ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D+ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O.      17. Billiards and Two-Dimensional Problems of Optimal Resistance      Alexander Plakhov 《Archive for Rational Mechanics and Analysis》2009,194(2):349-381 A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR2{B \subset \mathbb{R}^2} we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR2 \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.      18. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      19. Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales      S. H. Saker 《Nonlinear Oscillations》2011,13(3):407-428 Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation ( p(t)( [ y(t) + r(t)y( t(t) ) ]D )g )D + f( t,y( d(t) ) = 0,    t ? [ t0,￥ )\mathbbT, {\left( {p(t){{\left( {{{\left[ {y(t) + r(t)y\left( {\tau (t)} \right)} \right]}^\Delta }} \right)}^\gamma }} \right)^\Delta } + f\left( {t,y\left( {\delta (t)} \right)} \right. = 0,\quad t \in {\left[ {{t_0},\infty } \right)_\mathbb{T}},      20. On Kato’s Method for Navier–Stokes Equations      Bernhard H. Haak  Peer Chr. Kunstmann 《Journal of Mathematical Fluid Mechanics》2009,11(4):492-535 We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR3{\mathbb{R}}^{3} and irregular domains in \mathbbRn{\mathbb{R}}^{n}.      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

Symmetries of degenerate center singularities of plane vector fields

Let D² ì \mathbbR² {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ì \mathbbR² O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D⁺ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D ² that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D⁺ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j ¹ F(O) of F at O is nondegenerate. In this paper, the degenerate case j ¹ F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D⁺ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D⁺ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D⁺ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O.     

Billiards and Two-Dimensional Problems of Optimal Resistance

A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR²{B \subset \mathbb{R}^2} we assign a measure ν _B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR² \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν _B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.     

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales

Our aim is to establish some sufficient conditions for the oscillation of the second-order quasilinear neutral functional dynamic equation

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.