共查询到20条相似文献,搜索用时 812 毫秒
1.
Sarika Goyal K. Sreenadh 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(4):567-588
In this article, we study the Fu?ik spectrum of the fractional Laplace operator which is defined as the set of all \({(\alpha, \beta)\in \mathbb{R}^2}\) such that $$\quad \left.\begin{array}{ll}\quad (-\Delta)^s u = \alpha u^{+} - \beta u^{-} \quad {\rm in}\;\Omega \\ \quad \quad \quad u = 0 \quad \quad \quad \qquad {\rm in}\; \mathbb{R}^n{\setminus}\Omega.\end{array}\right\}$$ has a non-trivial solution u, where \({\Omega}\) is a bounded domain in \({\mathbb{R}^n}\) with Lipschitz boundary, n > 2s, \({s \in (0, 1)}\) . The existence of a first nontrivial curve \({\mathcal{C}}\) of this spectrum, some properties of this curve \({\mathcal{C}}\) , e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fu?ik spectrum. 相似文献
2.
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
3.
4.
We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C 2 boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ 1 and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω. 相似文献
5.
Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??). 相似文献
6.
Alano Ancona 《Mathematische Zeitschrift》2012,272(1-2):405-427
Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)2 is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V 0 ≤ V 1 and ${\mathcal{L}}$ a second order elliptic operator. 相似文献
7.
Huajun Gong Tobias Lamm Changyou Wang 《Calculus of Variations and Partial Differential Equations》2012,45(1-2):165-191
We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain ${\Omega\subset\mathbb R^n (n\ge 5)}$ to a compact, smooth Riemannian manifold ${N\subset{\mathbb {R}}^l}$ without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset ${\Sigma\subset\overline{\Omega}}$ , with ${H^{n-4}(\Sigma)=0}$ , such that ${\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}$ . 相似文献
8.
Grey Ercole 《Archiv der Mathematik》2014,103(2):189-194
We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution. 相似文献
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10.
Let G be a homogeneous group, and let X 1, X 2, … , X m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above. 相似文献
11.
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
12.
Samuel Littig Friedemann Schuricht 《Calculus of Variations and Partial Differential Equations》2014,49(1-2):707-727
Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ . 相似文献
13.
Fausto Acanfora 《Ricerche di matematica》2012,61(2):299-306
In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? h is a uniformly bounded sequence of connected open sets in R n , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L p , provided that W 1, ??(??) is dense in the space L 1, p (??) of all functions whose gradient belongs to L p (??, R n ). 相似文献
14.
Robin Nittka 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(3):1125-1155
We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all ${p \in (1,\infty)}$ , but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space ${\mathrm{C}(\overline{\Omega})}$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in ${\mathrm{C}(\overline{\Omega})}$ is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on ${\mathrm{C}(\overline{\Omega})}$ . 相似文献
15.
We establish interior gradient bounds for functions ${u \in W^1_{1, {\rm loc}} (\Omega)}$ which locally minimize the variational integral ${J [u, \Omega] = \int_\Omega h \left( |\nabla u| \right) dx}$ under the side condition ${u \ge \Psi}$ a.e. on Ω with obstacle ${\Psi}$ being locally Lipschitz. Here h denotes a rather general N-function allowing (p, q)-ellipticity with arbitrary exponents 1 < p ≤ q < ∞. Our arguments are based on ideas developed in Bildhauer et al. (Z Anal Anw 20:959–985, 2001) combined with techniques originating in Fuchs (2011). 相似文献
16.
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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18.
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
19.
S. Kračmar D. Medková Š. Nečasová W. Varnhorn 《Annali di Matematica Pura ed Applicata》2013,192(6):1059-1076
Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R 3. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ. 相似文献
20.
Michael Winkler 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(4):575-608
The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions ${{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}$ on the boundary of a bounded convex domain ${\Omega \subset {\mathbb{R}}^n}$ with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n?=?2, but previous mathematical results appear to concentrate on the case n?=?1. In this work, it is proved that when n??? 3,??? ?? 0, ???>?0 and ${f \in L^\infty(\Omega)}$ satisfies ${{\int_\Omega} f \ge 0}$ , for each prescribed initial distribution ${u_0 \in L^\infty(\Omega)}$ fulfilling ${{\int_\Omega} u_0 \ge 0}$ , there exists at least one global weak solution ${u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}$ satisfying ${{\int_\Omega} u(\cdot,t) \ge 0}$ for a.e. t?>?0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on??? and ${\|f\|_{L^\infty(\Omega)}}$ , it is shown that there exists a bounded set ${S\subset L^1(\Omega)}$ which is absorbing for ${(\star)}$ in the sense that for any such solution, we can pick T?>?0 such that ${e^{2\lambda u(\cdot,t)}\in S}$ for all t?>?T, provided that ?? is a ball and u 0 and f are radially symmetric with respect to x?=?0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional ${{\int_\Omega} e^{2\lambda u}dx}$ , but also the use of a maximum principle for second-order elliptic equations. 相似文献