Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn overdetermined problem associated to the Finsler Laplacian
311
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.
rate research
Read More
In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $omega$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for non-quasianalytic weight functions $omega$. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragmen-Lindelof principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.
Suppose $F: mathbb{R}^{N} rightarrow [0, +infty)$ be a convex function of class $C^{2}(mathbb{R}^{N} backslash {0})$ which is even and positively homogeneous of degree 1. We denote $gamma_1=inflimits_{uin W^{1, N}_{0}(Omega)backslash {0}}frac{int_{Omega}F^{N}( abla u)dx}{| u|_p^N},$ and define the norm $|u|_{N,F,gamma, p}=bigg(int_{Omega}F^{N}( abla u)dx-gamma| u|_p^Nbigg)^{frac{1}{N}}.$ Let $Omegasubset mathbb{R}^{N}(Ngeq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0leq gamma <gamma_1$, we have $$ sup_{uin W^{1, N}_{0}(Omega), |u|_{N,F,gamma, p}leq 1}int_{Omega}e^{lambda |u|^{frac{N}{N-1}}}dx<+infty, $$ where $0<lambda leq lambda_{N}=N^{frac{N}{N-1}} kappa_{N}^{frac{1}{N-1}}$ and $kappa_{N}$ is the volume of a unit Wulff ball. Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any $0 leqgamma <gamma_1$.
In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting $ss^{n-1}$ orthogonally. As an application, we show a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.
Let $H$ be a norm of ${bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Delta_H$ the Finsler-Laplace operator defined by $Delta_Hu:=mbox{div},(H( abla u) abla_xi H( abla u))$. In this paper we prove that the Finsler-Laplace operator $Delta_H$ acts as a linear operator to $H_0$-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation $$ partial_t u=Delta_H u,qquad xin{bf R}^N,quad t>0, $$ where $Nge 1$ and $partial_t:=partial/partial t$.
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the boundary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
