GWGeorg Sebastian Weiss
Georg Sebastian Weiss
*This is an algorithmically generated author record based on publicly available information.
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Georg Sebastian Weiss has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Simon Eberle
Basque Ctr Appl Math, Bilbao, Spain
9 shared publications - Henrik Shahgholian
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
4 shared publications - Alessio Figalli
Department of Mathematics, ETH Zürich, Zürich, Switzerland
2 shared publications - Mariana Smit Vega Garcia
Department of Mathematics, University of Duisburg-Essen, Essen, Germany
2 shared publications Reiner Schätzle
Universitat Tübingen, Germany
2 shared publications- Xavier Ros-Oton
The University of Texas at Austin
2 shared publications
Latest Publications
- Open Access
A min-max variational approach to the existence of gravity water waves
We establish the existence of gravity water waves by applying a mountain pass theorem to a singular perturbation of the Alt-Caffarelli functional associated with the two-dimensional water wave equations. Our approach is formulated entirely in physical coordinates and does not require the air phase to be connected, nor does it rely on symmetry or monotonicity in the $x$ or $y$ directions. The framework presented allows for both a variational approach to a variety of fluid equilibrium problems and for construction of min-max solutions to Bernoulli-type free boundary problems.Open AccessA min-max variational approach to the existence of gravity water waves
We establish the existence of gravity water waves by applying a mountain pass theorem to a singular perturbation of the Alt-Caffarelli functional associated with the two-dimensional water wave equations. Our approach is formulated entirely in physical coordinates and does not require the air phase to be connected, nor does it rely on symmetry or monotonicity in the $x$ or $y$ directions. The framework presented allows for both a variational approach to a variety of fluid equilibrium problems and for construction of min-max solutions to Bernoulli-type free boundary problems. - 22 July 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH
- p. 337-353
A singular perturbation approach to the Dirichlet-area minimisation problem
We study both one and two-phase minimisers of the Dirichlet-area energy $$ E(v) = \int\_{B\_1} |\nabla v|^2 + \mathrm{Per}({v>0},B\_1). $$ In the two-phase case, we show that the energies $$ E\_{\varepsilon}(v) = \int\_{B\_1}|\nabla v|^2 + \frac{1}{\varepsilon}W\Big(\frac{v}{\varepsilon^{1/2}}\Big), $$ $\Gamma$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $\[-1,1]$. As a consequence, we show that bounded local minimisers of $E\_{\varepsilon}$ converge to a local minimiser of $E$.- 22 July 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH
- p. 337-353
A singular perturbation approach to the Dirichlet-area minimisation problem
We study both one and two-phase minimisers of the Dirichlet-area energy $$ E(v) = \int\_{B\_1} |\nabla v|^2 + \mathrm{Per}({v>0},B\_1). $$ In the two-phase case, we show that the energies $$ E\_{\varepsilon}(v) = \int\_{B\_1}|\nabla v|^2 + \frac{1}{\varepsilon}W\Big(\frac{v}{\varepsilon^{1/2}}\Big), $$ $\Gamma$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $\[-1,1]$. As a consequence, we show that bounded local minimisers of $E\_{\varepsilon}$ converge to a local minimiser of $E$.- Top Cited Papers
- 1 January 2025
- Published by Annals of Mathematics in Annals of Mathematics
- Vol. 201 (1) , 167-224
Complete classification of global solutions to the obstacle problem
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied for more than 90 years. In this paper, we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.- 1 January 2025
- Published by Annals of Mathematics in Annals of Mathematics
- Vol. 201 (1) , 167-224
Top Cited PapersComplete classification of global solutions to the obstacle problem
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied for more than 90 years. In this paper, we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base. - Open Access
Classification of global solutions to the obstacle problem in the plane
Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions $n\geq 3$ was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.Open AccessClassification of global solutions to the obstacle problem in the plane
Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions $n\geq 3$ was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address. - Open Access
- 20 October 2023
- Published by Elsevier in Journal of Differential Equations
- Vol. 377, 873-887
The structure of the regular part of the free boundary close to singularities in the obstacle problem
- 20 October 2023
- Published by Elsevier in Journal of Differential Equations
- Vol. 377, 873-887
Open AccessThe structure of the regular part of the free boundary close to singularities in the obstacle problem
- Open Access
The structure of the regular part of the free boundary close to singularities in the obstacle problem
We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets in higher dimensions.
Open AccessThe structure of the regular part of the free boundary close to singularities in the obstacle problem
We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets in higher dimensions.
- Open Access
Complete classification of global solutions to the obstacle problem
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.Open AccessComplete classification of global solutions to the obstacle problem
The characterization of global solutions to the obstacle problems in $\mathbb{R}^N$, or equivalently of null quadrature domains, has been studied over more than 90 years. In this paper we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base. - Open Access
On global solutions of the obstacle problem
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.
Open AccessOn global solutions of the obstacle problem
Assuming a lower bound on the dimension, we prove a long standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.
- 9 July 2021
- Published by American Mathematical Society (AMS) in St. Petersburg Mathematical Journal
- Vol. 32 (4) , 705-711
Characterizing compact coincidence sets in the obstacle problem—a short proof
The objective of this article is to present a concise and easy-to-extend proof of the known fact that coincidence sets of global solutions of the obstacle problem that are bounded and have nonempty interior are ellipsoids.
- 9 July 2021
- Published by American Mathematical Society (AMS) in St. Petersburg Mathematical Journal
- Vol. 32 (4) , 705-711
Characterizing compact coincidence sets in the obstacle problem—a short proof
The objective of this article is to present a concise and easy-to-extend proof of the known fact that coincidence sets of global solutions of the obstacle problem that are bounded and have nonempty interior are ellipsoids.
- 21 June 2021
- Published by Elsevier in Nonlinear Analysis
- Vol. 211, 112473
Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian
- 21 June 2021
- Published by Elsevier in Nonlinear Analysis
- Vol. 211, 112473
Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian