共查询到20条相似文献,搜索用时 31 毫秒
1.
Atsushi Tachikawa 《Calculus of Variations and Partial Differential Equations》2014,50(1-2):145-169
We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ . 相似文献
2.
In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$ 相似文献
3.
We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given. 相似文献
4.
Pablo Figueroa 《Calculus of Variations and Partial Differential Equations》2014,49(1-2):613-647
We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ . 相似文献
5.
Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e 1, e 2, . . . , e n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ . 相似文献
6.
B. Canuto 《Calculus of Variations and Partial Differential Equations》2014,50(1-2):305-334
Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions. 相似文献
7.
Paolo Caldiroli Roberta Musina 《Calculus of Variations and Partial Differential Equations》2012,45(1-2):147-164
Let Ω be a cone in ${\mathbb {R}^{n}}$ with n ≥? 2. For every fixed ${\alpha \in \mathbb {R}}$ we find the best constant in the Rellich inequality ${\int\nolimits_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx \ge C\int\nolimits_{\Omega}|x|^{\alpha-4}|u|^{2}dx}$ for ${u \in C^{2}_{c}(\overline\Omega\setminus\{0\})}$ . We also estimate the best constant for the same inequality on ${C^{2}_{c}(\Omega)}$ . Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains. 相似文献
8.
Huiqiang Jiang Christopher J. Larsen Luis Silvestre 《Calculus of Variations and Partial Differential Equations》2011,42(3-4):301-321
Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P ??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??. 相似文献
9.
Jun Cao Der-Chen Chang Dachun Yang Sibei Yang 《Integral Equations and Operator Theory》2013,76(2):225-283
Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H ∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ . 相似文献
10.
Salomón Alarcón Leonelo Iturriaga Alexander Quaas 《Calculus of Variations and Partial Differential Equations》2012,45(3-4):443-454
We study the problem $$ \left\{\begin{array}{ll} {-\varepsilon^{2}\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u) = f (x, u)} \quad\; {\rm in} \; \Omega,\\ {u = 0} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {\rm on} \; \partial{\Omega}, \end{array} \right.$$ where Ω is a smooth bounded domain in ${\mathbb{R}^{N},N > 2,}$ and show it possesses nontrivial solutions for small values of ε provided f is a nonnegative continuous function which has a positive zero. The multiplicity result is based on degree theory together with a new Liouville type theorem for ${-{M}^+_{\lambda,\Lambda}(D^{2}u) = f(u)}$ in ${\mathbb{R}^{N}}$ for nonnegative nonlinearities with zeros. 相似文献
11.
Diego Moreira Lihe Wang 《Calculus of Variations and Partial Differential Equations》2014,49(3-4):1237-1261
In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained. 相似文献
12.
Johannes Lankeit Patrizio Neff Dirk Pauly 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(6):1679-1688
Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ . 相似文献
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15.
Giuseppe Maria Coclite Gisèle Ruiz Goldstein Jerome A. Goldstein 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2013,64(3):733-753
Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$\left\{ \begin{array}{lll} u_{tt} ={\rm div} (\mathcal{A} \nabla u)-\gamma (x,u_t), && \quad {\rm in} \; (0, \infty) \times \Omega,\\ u(0, \cdot)=f, \, u_t(0,\cdot)=g, && \quad {\rm in}\; \Omega, \\ u_{tt} + \beta \partial^ \mathcal{A}_\nu u+c(x)u+ \delta (x,u_t)-q \beta \Lambda_{\rm LB} u=0,&& \quad {\rm on} \;(0, \infty ) \times \partial \Omega . \end{array}\right. $$ for t ≥ 0 and ${x \in \Omega \subset \mathbb{R}^N}$ ; the last equation holds on the boundary ?Ω. Here ${\mathcal{A}= \{a_{ij}(x)\}_{ij}}$ is a real, hermitian, uniformly positive definite N × N matrix; ${\beta \in C(\partial \Omega)}$ , with β > 0; ${\gamma:\Omega \times \mathbb{R} \to \mathbb{R}; \delta:\partial \Omega \times \mathbb{R} \to \mathbb{R}; \,c:\partial \Omega \to \mathbb{R}; \, q \ge 0, \Lambda_{\rm LB}}$ is the Laplace–Beltrami operator on ?Ω, and ${\partial^\mathcal{A}_\nu u}$ is the conormal derivative of u with respect to ${\mathcal{A}}$ ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients ${\mathcal{A},\,\beta,\,\gamma,\,\delta,\,c,\,q}$ , and the initial conditions f, g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q. 相似文献
16.
Lubomira G. Softova 《manuscripta mathematica》2011,136(3-4):365-382
We consider regular oblique derivative problem in cylinder Q T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem. 相似文献
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18.
Luigi Santocanale 《Algebra Universalis》2010,63(2-3):101-130
For L a finite lattice, let ${\mathbb {C}(L) \subseteq L^2}$ denote the set of pairs γ = (γ 0, γ 1) such that ${\gamma_0 \prec \gamma_1}$ and order it as followsγ ≤ δ iff γ 0 ≤ δ 0, ${\gamma_{1} \nleq \delta_0,}$ and γ 1 ≤ δ 1. Let ${\mathbb {C}(L, \gamma)}$ denote the connected component of γ in this poset. Our main result states that, for any ${\gamma, \mathbb {C}(L, \gamma)}$ is a semidistributive lattice if L is semidistributive, and that ${\mathbb {C}(L, \gamma)}$ is a bounded lattice if L is bounded. Let ${\mathcal{S}_{n}}$ be the Permutohedron on n letters and let ${\mathcal{T}_{n}}$ be the Associahedron on n + 1 letters. Explicit computations show that ${\mathbb {C}(\mathcal{S}_{n}, \alpha) = \mathcal{S}_{n-1}}$ and ${\mathbb {C}(\mathcal {T}_n, \alpha) = \mathcal {T}_{n-1}}$ , up to isomorphism, whenever α1 is an atom of ${\mathcal{S}_{n}}$ or ${\mathcal{T}_{n}}$ . These results are consequences of new characterizations of finite join-semidistributive and of finite lower bounded lattices: (i) a finite lattice is join-semidistributive if and only if the projection sending ${\gamma \in \mathbb {C}(L)}$ to ${\gamma_0 \in L}$ creates pullbacks, (ii) a finite join-semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are a generalization of the tools used by Caspard et al. to prove that lattices of finite Coxeter groups are bounded. 相似文献
19.
Luís V. Pessoa 《Complex Analysis and Operator Theory》2013,7(5):1569-1581
Let $k$ and $j$ be positive integers. We prove that the action of the two-dimensional singular integral operators $(S_\Omega )^{j-1}$ and $(S_\Omega ^*)^{j-1}$ on a Hilbert base for the Bergman space $\mathcal{A }^2(\Omega )$ and anti-Bergman space $\mathcal{A }^2_{-1}(\Omega ),$ respectively, gives Hilbert bases $\{ \psi _{\pm j , k } \}_{ k }$ for the true poly-Bergman spaces $\mathcal{A }_{(\pm j)}^2(\Omega ),$ where $S_\Omega $ denotes the compression of the Beurling transform to the Lebesgue space $L^2(\Omega , dA).$ The functions $\psi _{\pm j,k}$ will be explicitly represented in terms of the $(2,1)$ -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of $\Omega $ . We establish Rodrigues type formulas for the poly-Bergman kernels of $\mathbb{D }$ . 相似文献
20.
L. H. de Miranda M. Montenegro 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(5):1683-1699
In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ . 相似文献