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Register a new userStability estimates in $H^1_0$ for solutions of elliptic equations in varying domains
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We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $Omega$ and on the domain $phi(Omega)$ resulting from $Omega$ by means of a bi-Lipschitz map $phi$. We consider the solutions $u$ and $tilde u$ of the corresponding elliptic equations with the same right-hand side $fin L^2(Omegacupphi(Omega))$. Under certain assumptions we estimate the difference $| ablatilde u- abla u|_{L^2(Omegacupphi(Omega))}$ in terms of certain measure of vicinity of $phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $phi(Omega)$ and $Omega$, that is $|phi(Omega)triangle Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.
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We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $phi (Omega)$ parametrized by Lipschitz homeomorphisms $phi $ defined on a fixed reference domain $Omega$. Given two open sets $phi (Omega)$, $tilde phi (Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $|tilde phi -phi |_{W^{1,p}(Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.
In this paper, we show $C^{2,alpha}$ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.
This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^infty$-norm for the solutions gradient in terms of its local $L^p$-norm. Specifically, we prove begin{equation*} | abla u|_{L^infty(B_{frac{R}{2}}(x_0))}^p leq frac{C}{|B_R(x_0)|}int_{B_R(x_0)}| abla u(x)|^p dx. end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into $G$-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.
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