Friedemann Brock
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View article: Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights Open
We study the following class of Steklov eigenvalue problems: \[ \nabla \cdot \bigl( w \nabla u \bigr) = 0 \quad \text{in } Ω, \qquad \frac{\partial u}{\partial ν} = γv u \quad \text{on } \partial Ω, \] where $w$ and $v$ are prescribed posi…
View article: On the reverse isoperimetric inequality in Gauss space
On the reverse isoperimetric inequality in Gauss space Open
In this paper, we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in \mathbb{R}^{2} . While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects …
View article: On the reverse isoperimetric inequality in Gauss space
On the reverse isoperimetric inequality in Gauss space Open
In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects …
View article: Shape of extremal functions for weighted Sobolev-type inequalities
Shape of extremal functions for weighted Sobolev-type inequalities Open
We study the shape of solutions to certain variational problems in Sobolev spaces with weights that are powers of ∣ x ∣ | x| . In particular, we detect situations when the extremal functions lack symmetry properties such as radial symmet…
View article: Shape of extremal functions for weighted Sobolev-type inequalities
Shape of extremal functions for weighted Sobolev-type inequalities Open
We study the shape of solutions to some variational problems in Sobolev spaces with weights that are powers of |x|. In particular, we detect situations when the extremal functions lack symmetry properties such as radial symmetry and antisy…
View article: Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball
Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball Open
We consider non-negative distributional solutions to the equation in a ball , with on , where is continuous and non-increasing in the first variable and , with and for . According to a result of the first author, the solutions satisf…
View article: Steiner symmetrization for anisotropic quasilinear equations \nvia partial discretization
Steiner symmetrization for anisotropic quasilinear equations
\nvia partial discretization Open
View article: Some isoperimetric inequalities with respect to monomial weights
Some isoperimetric inequalities with respect to monomial weights Open
We solve a class of isoperimetric problems on ℝ + 2 with respect to monomial weights. Let α and β be real numbers such that 0 ≤ α < β + 1, β ≤ 2 α . We show that, among all smooth sets Ω in ℝ + 2 with fixed weighted measure ∬ Ω y β d x d y…
View article: Steiner symmetrization for anisotropic quasilinear equations via partial discretization
Steiner symmetrization for anisotropic quasilinear equations via partial discretization Open
In this paper we obtain comparison results for the quasilinear equation −\mathrm{\Delta }_{p,x}u−u_{yy} = f with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable x , thus solving a long open problem. In fact, …
View article: Some Weighted Isoperimetric Problems on $${\mathbb {R}}^N _+ $$ with Stable Half Balls Have No Solutions
Some Weighted Isoperimetric Problems on $${\mathbb {R}}^N _+ $$ with Stable Half Balls Have No Solutions Open
View article: Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball
Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball Open
We consider non-negative distributional solutions $u\in C^1 (\bar{B_R } )$ to the equation $-\mbox{div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$ in a ball $B_R$, with $u=0$ on $\partial B_R $, where $f$ is continuous and non-in…
View article: The isoperimetric problem for a class of non-radial weights and applications
The isoperimetric problem for a class of non-radial weights and applications Open
View article: A unified approach to symmetry for semilinear equations associated to the Laplacian in $\mathbb{R}^N$
A unified approach to symmetry for semilinear equations associated to the Laplacian in $\mathbb{R}^N$ Open
We show radial symmetry of positive solutions to the Hénon equation $-Δu = |x|^{-\ell} u^q $ in $\mathbb{R}^N \setminus \{ 0\} $, where $\ell \geq 0$, $q>0$ and satisfy further technical conditions. A new ingredient is a maximum principle …
View article: Some weighted isoperimetric problems on $\\mathbb{R}^N _+ $ with stable\n half balls have no solutions
Some weighted isoperimetric problems on $\\mathbb{R}^N _+ $ with stable\n half balls have no solutions Open
We show the counter-intuitive fact that some weighted isoperimetric problems\non the half-space $ \\mathbb{R}^N _+ $, for which half-balls centered at the\norigin are stable, have no solutions. A particular case is the measure $d\\mu =\nx_…
View article: The isoperimetric problem for a class of non-radial weights and applications
The isoperimetric problem for a class of non-radial weights and applications Open
We study a class of isoperimetric problems on R+N where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type |x|kxNα. Our results imply some sharp functional inequalities, …
View article: On weighted isoperimetric inequalities with non-radial densities
On weighted isoperimetric inequalities with non-radial densities Open
We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^α$. We solve them in a special case while a more detailed study is contained in …
View article: New Pólya–Szegö-type inequalities and an alternative approach to comparison results for PDE's
New Pólya–Szegö-type inequalities and an alternative approach to comparison results for PDE's Open
View article: The isoperimetric problem for a class of non-radial weights and\n applications
The isoperimetric problem for a class of non-radial weights and\n applications Open
We study a class of isoperimetric problems on $\\mathbb{R}^{N}_{+} $ where the\ndensities of the weighted volume and weighted perimeter are given by two\ndifferent non-radial functions of the type $|x|^k x_N^\\alpha$. Our results\nimply so…
View article: Symmetry and asymmetry of minimizers of a class of noncoercive functionals
Symmetry and asymmetry of minimizers of a class of noncoercive functionals Open
In this paper we prove symmetry results for minimizers of a noncoercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e., they are axially symmetri…
View article: New P\\'olya-Szeg\\"o-type inequalities and an alternative approach to\n comparison results for PDE's
New P\\'olya-Szeg\\"o-type inequalities and an alternative approach to\n comparison results for PDE's Open
We prove some P\\'olya-Szeg\\"o type inequalities which involve couples of\nfunctions and their rearrangements. Our inequalities reduce to the classical\nP\\'olya-Szeg\\"o principle when the two functions coincide. As an application,\nwe g…
View article: Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $
Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $ Open
We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fix…
View article: Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^α$
Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^α$ Open
We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $Ω$ in $\mathbb{R} ^N$ with fixed Le…
View article: Symmetry for a general class of overdetermined elliptic problems
Symmetry for a general class of overdetermined elliptic problems Open
View article: An isoperimetric inequality for Gauss-like product measures
An isoperimetric inequality for Gauss-like product measures Open
View article: Symmetry and asymmetry of minimizers of a class of noncoercive functionals
Symmetry and asymmetry of minimizers of a class of noncoercive functionals Open
In this paper we prove symmetry results for minimizers of a non coercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e. they are axially symmetri…
View article: Optimal Szegö-Weinberger type inequalities
Optimal Szegö-Weinberger type inequalities Open
Denote with $\mu _{1}(\Omega ;e^{h( |x|) })$ the first nontrivialeigenvalue of the Neumann problem\begin{eqnarray}&-div( e^{h( |x|) }\nabla u) =\mu e^{h(|x|) }u \quad in \ \Omega \\&\frac{\partial u}{\partial \nu }=0 \quad on \ \partial \O…
View article: Symmetry for a general class of overdetermined elliptic problems
Symmetry for a general class of overdetermined elliptic problems Open
Let $Ω$ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{Ω}) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $Ω$ and $u(x)=0, \ |\nabla …