Search results for " 35J70"
showing 7 items of 7 documents
Uniqueness of diffusion on domains with rough boundaries
2016
Let $\Omega$ be a domain in $\mathbf R^d$ and $h(\varphi)=\sum^d_{k,l=1}(\partial_k\varphi, c_{kl}\partial_l\varphi)$ a quadratic form on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the $c_{kl}$ are real symmetric $L_\infty(\Omega)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in \Omega$. Further assume there are $a, \delta>0$ such that $a^{-1}d_\Gamma^{\delta}\,I\le C\le a\,d_\Gamma^{\delta}\,I$ for $d_\Gamma\le 1$ where $d_\Gamma$ is the Euclidean distance to the boundary $\Gamma$ of $\Omega$. We assume that $\Gamma$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $\Gamma$, is larger or equal to $d-1$ we also assume a mild uniformity property for $\Omega$ i…
A weak comparison principle for solutions of very degenerate elliptic equations
2012
We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where \(\nabla u\in K\), where \(K\subset \mathbb{R }^N\) is a Borel set containing the origin.
An optimal Poincaré-Wirtinger inequality in Gauss space
2013
International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.
The method of moving planes: a quantitative approach
2018
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.
Symmetry of minimizers with a level surface parallel to the boundary
2015
We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differenti…
C1,α-regularity for variational problems in the Heisenberg group
2017
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.
The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces
2020
In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expressio…