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Tobias Weth - One of the best experts on this subject based on the ideXlab platform.
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serrin s Overdetermined Problem on the sphere
Calculus of Variations and Partial Differential Equations, 2018Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study Serrin’s Overdetermined boundary value Problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial \Omega \end{aligned}$$ in subdomains $$\Omega $$ of the round unit sphere $$S^N \subset \mathbb {R}^{N+1}$$ , where $$\Delta _{S^N}$$ denotes the Laplace–Beltrami operator on $$S^N$$ . A subdomain $$\Omega $$ of $$S^N$$ is called a Serrin domain if it admits a solution of this Overdetermined Problem. In our main result, we construct Serrin domains in $$S^N$$ , $$N \ge 2$$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $$S^{N}$$ which are not bounded by geodesic spheres.
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unbounded periodic solutions to serrin s Overdetermined boundary value Problem
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study the existence of nontrivial unbounded domains \({\Omega}\) in \({{\mathbb R}^{N}}\) such that the Overdetermined Problem $${-\Delta u = 1 \quad {\rm in} \, \Omega}, \quad u = 0, \quad \partial_{\nu} u = {\rm const} \quad {\rm on} \partial \Omega$$ admits a solution u. By this, we complement Serrin’s classification result from 1971, which yields that every bounded domain admitting a solution of the above Problem is a ball in \({{\mathbb R}^{N}}\). The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the Problem.
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unbounded periodic solutions to serrin s Overdetermined boundary value Problem
arXiv: Analysis of PDEs, 2016Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study the existence of nontrivial unbounded domains $\Omega$ in $\mathbb{R}^N$ such that the Overdetermined Problem $$ -\Delta u = 1 \quad \text{in $\Omega$}, \qquad u=0, \quad \partial_\nu u=\textrm{const} \qquad \text{on $\partial \Omega$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above Problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the Problem.
Rolando Magnanini - One of the best experts on this subject based on the ideXlab platform.
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nearly optimal stability for serrin s Problem and the soap bubble theorem
Calculus of Variations and Partial Differential Equations, 2020Co-Authors: Rolando Magnanini, Giorgio PoggesiAbstract:We present new quantitative estimates for the radially symmetric configuration concerning Serrin’s Overdetermined Problem for the torsional rigidity, Alexandrov’s Soap Bubble theorem, and other related Problems. The new estimates improve on those obtained in Magnanini and Poggesi (J Anal Math, 139(1), 179–205, 2019), Magnanini and Poggesi (Indiana Univ Math J, arXiv:1708.07392, 2017) and are in some cases optimal.
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serrin s Problem and alexandrov s soap bubble theorem enhanced stability via integral identities
arXiv: Analysis of PDEs, 2017Co-Authors: Rolando Magnanini, Giorgio PoggesiAbstract:We consider Serrin's Overdetermined Problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two Problems and help to obtain better (in some cases optimal) quantitative estimates for the radially symmetric configuration. The estimates for the Soap Bubble Theorem benefit from those of Serrin's Problem.
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holder stability for serrin s Overdetermined Problem
Annali di Matematica Pura ed Applicata, 2016Co-Authors: Giulio Ciraolo, Rolando Magnanini, Vincenzo VespriAbstract:In a bounded domain \(\varOmega \), we consider a positive solution of the Problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–932, 1999) for Serrin’s Overdetermined Problem. It also extends to a large class of semilinear equations the Holder estimate obtained in Brandolini et al. (J Differ Equ 245:1566–1583, 2008) for the case of torsional rigidity (\(f\equiv 1\)) by means of integral identities. The proof hinges on ideas contained in Aftalion et al. (1999) and uses Carleson-type estimates and improved Harnack inequalities in cones.
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symmetry and linear stability in serrin s Overdetermined Problem via the stability of the parallel surface Problem
arXiv: Analysis of PDEs, 2015Co-Authors: Giulio Ciraolo, Rolando Magnanini, Vincenzo VespriAbstract:We consider the solution of the Problem �u = f(u) and u > 0 in , u = 0 on , where is a bounded domain in R N with boundary of class C 2,� , 0 0, where re and ri are the radii of a spherical annulus containing , � is a surface parallel to at distance � and sufficiently close to , and ( u) � is the Lipschitz semi-norm of u on � ; secondly, if in addition uis constant on , show that (u) � = o(C�) as � ! 0 + . In this paper, we prove that this strategy is successful. As a by-product of this method, for C 2,� -regular domains, we also obtain a linear stability estimate for Serrin's symmetry result. Our result is optimal and greatly improves the similar logarithmic-type estimate of (ABR) and the Holder estimate of (CMV) that was restricted to convex domains.
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h older stability for serrin s Overdetermined Problem
arXiv: Analysis of PDEs, 2014Co-Authors: Giulio Ciraolo, Rolando Magnanini, Vincenzo VespriAbstract:In a bounded domain $\Omega$, we consider a positive solution of the Problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $\Omega$ (for instance, if $\Omega$ is convex), we show that $\partial\Omega$ is contained in a spherical annulus of radii $r_i 0$ and $\alpha\in (0,1]$. Here, $[u_\nu]_{\partial\Omega}$ is the Lipschitz seminorm on $\partial\Omega$ of the normal derivative of $u$. This result improves to H\"older stability the logarithmic estimate obtained in [1] for Serrin's Overdetermined Problem. It also extends to a large class of semilinear equations the H\"older estimate obtained in [6] for the case of torsional rigidity ($f\equiv 1$) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.
Giulio Ciraolo - One of the best experts on this subject based on the ideXlab platform.
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an exterior Overdetermined Problem for finsler n laplacian in convex cones
arXiv: Analysis of PDEs, 2021Co-Authors: Giulio CiraoloAbstract:We consider a partially Overdetermined Problem for anisotropic $N$-Laplace equations in a convex cone $\Sigma$ intersected with the exterior of a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $\Sigma\cap\Omega$ must be the intersection of the Wulff shape and $\Sigma$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.
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on serrin s Overdetermined Problem in space forms
Manuscripta Mathematica, 2019Co-Authors: Giulio Ciraolo, Luigi VezzoniAbstract:We consider Serrin’s Overdetermined Problem for the equation $$\Delta v + nK v = -\,1$$ in space forms, where K is the curvature of the space, and we prove a symmetry result by using a P-function approach. Our approach generalizes the one introduced by Weinberger to space forms and, as in the Euclidean case, it provides a short proof of the symmetry result which does not make use of the method of moving planes.
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on serrin s Overdetermined Problem in space forms
arXiv: Analysis of PDEs, 2017Co-Authors: Giulio Ciraolo, Luigi VezzoniAbstract:We consider an Overdetermined Serrin's type Problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function.
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holder stability for serrin s Overdetermined Problem
Annali di Matematica Pura ed Applicata, 2016Co-Authors: Giulio Ciraolo, Rolando Magnanini, Vincenzo VespriAbstract:In a bounded domain \(\varOmega \), we consider a positive solution of the Problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–932, 1999) for Serrin’s Overdetermined Problem. It also extends to a large class of semilinear equations the Holder estimate obtained in Brandolini et al. (J Differ Equ 245:1566–1583, 2008) for the case of torsional rigidity (\(f\equiv 1\)) by means of integral identities. The proof hinges on ideas contained in Aftalion et al. (1999) and uses Carleson-type estimates and improved Harnack inequalities in cones.
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An Overdetermined Problem for the anisotropic capacity
Calculus of Variations and Partial Differential Equations, 2016Co-Authors: Chiara Bianchini, Giulio Ciraolo, Paolo SalaniAbstract:We consider an Overdetermined Problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an Overdetermined Problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).
Mouhamed Moustapha Fall - One of the best experts on this subject based on the ideXlab platform.
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serrin s Overdetermined Problem on the sphere
Calculus of Variations and Partial Differential Equations, 2018Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study Serrin’s Overdetermined boundary value Problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial \Omega \end{aligned}$$ in subdomains $$\Omega $$ of the round unit sphere $$S^N \subset \mathbb {R}^{N+1}$$ , where $$\Delta _{S^N}$$ denotes the Laplace–Beltrami operator on $$S^N$$ . A subdomain $$\Omega $$ of $$S^N$$ is called a Serrin domain if it admits a solution of this Overdetermined Problem. In our main result, we construct Serrin domains in $$S^N$$ , $$N \ge 2$$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $$S^{N}$$ which are not bounded by geodesic spheres.
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unbounded periodic solutions to serrin s Overdetermined boundary value Problem
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study the existence of nontrivial unbounded domains \({\Omega}\) in \({{\mathbb R}^{N}}\) such that the Overdetermined Problem $${-\Delta u = 1 \quad {\rm in} \, \Omega}, \quad u = 0, \quad \partial_{\nu} u = {\rm const} \quad {\rm on} \partial \Omega$$ admits a solution u. By this, we complement Serrin’s classification result from 1971, which yields that every bounded domain admitting a solution of the above Problem is a ball in \({{\mathbb R}^{N}}\). The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the Problem.
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unbounded periodic solutions to serrin s Overdetermined boundary value Problem
arXiv: Analysis of PDEs, 2016Co-Authors: Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias WethAbstract:We study the existence of nontrivial unbounded domains $\Omega$ in $\mathbb{R}^N$ such that the Overdetermined Problem $$ -\Delta u = 1 \quad \text{in $\Omega$}, \qquad u=0, \quad \partial_\nu u=\textrm{const} \qquad \text{on $\partial \Omega$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above Problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the Problem.
Giorgio Poggesi - One of the best experts on this subject based on the ideXlab platform.
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quantitative stability estimates for a two phase serrin type Overdetermined Problem
arXiv: Analysis of PDEs, 2021Co-Authors: Lorenzo Cavallina, Giorgio Poggesi, Toshiaki YachimuraAbstract:In this paper, we deal with an Overdetermined Problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase Overdetermined Problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase Problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner Problem by the aforementioned two-phase stability result.
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nearly optimal stability for serrin s Problem and the soap bubble theorem
Calculus of Variations and Partial Differential Equations, 2020Co-Authors: Rolando Magnanini, Giorgio PoggesiAbstract:We present new quantitative estimates for the radially symmetric configuration concerning Serrin’s Overdetermined Problem for the torsional rigidity, Alexandrov’s Soap Bubble theorem, and other related Problems. The new estimates improve on those obtained in Magnanini and Poggesi (J Anal Math, 139(1), 179–205, 2019), Magnanini and Poggesi (Indiana Univ Math J, arXiv:1708.07392, 2017) and are in some cases optimal.
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radial symmetry for p harmonic functions in exterior and punctured domains
arXiv: Analysis of PDEs, 2018Co-Authors: Giorgio PoggesiAbstract:We prove symmetry for the p-capacitary potential satisfying $$ \Delta_p u = 0 \, \mbox{ in } \mathbb{R}^N \setminus \overline{\Omega} , \; u=1 \, \mbox{ on } \Gamma, \; \lim_{|x|\rightarrow \infty} u(x)=0 , \; \; \; \; \; \; \; \; 1
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serrin s Problem and alexandrov s soap bubble theorem enhanced stability via integral identities
arXiv: Analysis of PDEs, 2017Co-Authors: Rolando Magnanini, Giorgio PoggesiAbstract:We consider Serrin's Overdetermined Problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two Problems and help to obtain better (in some cases optimal) quantitative estimates for the radially symmetric configuration. The estimates for the Soap Bubble Theorem benefit from those of Serrin's Problem.
