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Register a new userFredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions
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We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
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Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOmegasetminus{0}$. We prove that if $frac{N+1}{N-1} < q < frac{N+2}{N-2}$, then $u(x)leq C abs{x}^{-frac{2}{q-1}}$ and we compute the limit of $abs{x}^{frac{2}{q-1}} u(x)$ as $x to 0$. We also investigate the case $q= frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
Let $Omega subset mathbb{R}^N$ be a bounded domain and $delta(x)$ be the distance of a point $xin Omega$ to the boundary. We study the positive solutions of the problem $Delta u +frac{mu}{delta(x)^2}u=u^p$ in $Omega$, where $p>0, ,p e 1$ and $mu in mathbb{R},,mu e 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $delta^{- frac2{p-1}}$ at the boundary. In general domains we extend results of arXiv:arch-ive/1407.0288 and show that there exists a unique singular solutions $u$ such that $u/delta^{beta_-}to c$ on the boundary for an arbitrary positive function $c in C^{2+gamma}(partialOmega) , (gamma in (0,1)), c ge 0$. Here $beta_-$ is the smaller root of $beta(beta-1)+mu=0$.
We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces by restriction a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $pi_alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $Gamma$ is finite abelian. We prove then that the restriction $pi_alpha(P)$ is Fredholm if, and only if, $P$ is $alpha$-elliptic, a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $alpha$-elliptic, but the converse is not true in general. However, if $Gamma$ acts freely on a dense open subset of $M$, then $P$ is $alpha$-elliptic for the given fixed $alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $psi^{m}(M; E)^Gamma$ of classical, $Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E to M$ and of the structure of its restrictions to the isotypical components of $Gamma$. These structures are described in terms of the isotropy groups of the action of the group $Gamma$ on $E to M$.
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