arXiv subjects

Georg S. Weiss

Publications and source records attributed to Georg S. Weiss.

At least 19 recordsLinked to original sources

arXiv2026

Complete Rigidity at infinity and Existence of the Levinson Cavity

We present a potential theoretic approach reducing the analysis of the asymptotic shape of free surfaces to the analysis of a precise ordinary differential equation resulting from the reduction process. Although the approach relies mainly on the principal part of the PDE operator to allow for a representation formula and is thus not restricted to problems of elliptic type, we present it at the clean-cut example of three-dimensional axially symmetric steady incompressible cavity flows, which are Neumann-type Bernoulli free boundary problems and for which frequency formulas are unknown and, if they do exist, insufficient to yield the very precise asymptotic behavior we prove here. In 1946 Norman Levinson derived by a power-law ansatz with a slowly varying correction a precise formula for the asymptotic shape of such cavities. However his result requires very strong assumptions such that it has remained an open problem for 80 years whether the cavity solutions we know to exist by a result by Garabedian-Lewy-Schiffer [12] actually share this asymptotic behavior, or whether at least one solution possessing the Levinson asymptotics exists. Here we answer both questions affirmatively, and we obtain complete rigidity at infinity of the Levinson solution in the class of axially symmetric solutions, that is, any solution satisfying mild and natural assumptions at the fixed boundary and infinity converges asymptotically to the Levinson profile $(\log r)^{-1/4}\sqrt{r}$.

math.AP↗

arXiv2026

ACF Almost Monotonicity at Infinity with Applications to Perturbed Global Solutions

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we prove that, locally around regular points sufficiently far out, the cross-sections of $\{u=0\}$ perpendicular to $e_n$ are $C^2$ perturbations of ellipsoids. The main ingredient is a new large-scale almost monotonicity formula for the Alt--Caffarelli--Friedman functional. In contrast with the classical small-scale perturbative theory, our argument exploits the stability of the obstacle problem together with the fact that local perturbations vanish under blow-down. The method provides a model mechanism for controlling errors at infinity in stable free boundary problems.

math.AP↗

arXiv2025

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.

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arXiv2024

Rectifiability, finite Hausdorff measure, and compactness for non-minimizing Bernoulli free boundaries

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time-dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points $u$ -- which can be obtained as limits of classical solutions or limits of a singular perturbation problem -- it has been open since [Weiss03] whether the singular set can be large and what equation the measure $Δu$ satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber-Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary $\partial \{u > 0\}$ (including degenerate singular points $x$, at which $u(x + r \cdot)/r \rightarrow 0$ as $r\rightarrow 0$) is countably $\mathcal{H}^{n-1}$-rectifiable and has locally finite $\mathcal{H}^{n-1}$-measure, and we identify the measure $Δu$ completely. This gives a more precise characterization of the free boundary of $u$ in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

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arXiv2024

A singular perturbation approach to the Dirichlet-area minimisation problem

We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v) = \int_{B_1} \vert\nabla v\vert^2 + Per(\{v>0\},B_1).$$ In the two-phase case, we show that the energies $$E_{\varepsilon}(v) = \int_{B_1}\vert\nabla v\vert^2 + \frac{1}{\varepsilon}W\left(\frac{v}{\varepsilon^{1/2}}\right),$$ $Γ$-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 $E$.

math.AP↗

arXiv2022

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.

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arXiv2022

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.

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arXiv2021

Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation

The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially relevant procedure. Till this day in the scientific literature there is in general no mathematical convergence analysis which explains the numerical success of GD type optimization schemes in the training of ANNs with ReLU activation. GD type optimization schemes can be regarded as temporal discretization methods for the gradient flow (GF) differential equations associated to the considered optimization problem and, in view of this, it seems to be a natural direction of research to first aim to develop a mathematical convergence theory for time-continuous GF differential equations and, thereafter, to aim to extend such a time-continuous convergence theory to implementable time-discrete GD type optimization methods. In this article we establish two basic results for GF differential equations in the training of fully-connected feedforward ANNs with one hidden layer and ReLU activation. In the first main result of this article we establish in the training of such ANNs under the assumption that the probability distribution of the input data of the considered supervised learning problem is absolutely continuous with a bounded density function that every GF differential equation admits for every initial value a solution which is also unique among a suitable class of solutions. In the second main result of this article we prove in the training of such ANNs under the assumption that the target function and the density function of the probability distribution of the input data are piecewise polynomial that every non-divergent GF trajectory converges with an appropriate rate of convergence to a critical point and that the risk of the non-divergent GF trajectory converges with rate 1 to the risk of the critical point.

cs.LG↗

arXiv2021

Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian

In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N \geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.

math.AP↗

arXiv2021

Regularity of the free boundary for a parabolic cooperative system

In this paper we study the following parabolic system \begin{equation*} Δ\u -\partial_t \u =|\u|^{q-1}\u\,χ_{\{ |\u|>0 \}}, \qquad \u = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|\u | >0\}$. For $0\leq q<1$, we prove optimal growth rate for solutions $\u $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1, α}$ in space directions and half-Lipschitz in the time direction.

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arXiv2020

Characterizing compact coincidence sets in the obstacle problem -- a short proof

Motivated by the almost completely open problem of characterizing unbounded coincidence sets of global solutions of the classical obstacle problem in higher dimensions, we give in this note a concise and easy-to-extend proof of the known fact that if the coincidence set $\{u=0 \}$ of a global solution $u$ is bounded with nonempty interior then it is an ellipsoid.

math.AP↗

arXiv2016

Singularities in axisymmetric free boundaries for ElectroHydroDynamic equations

We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new.

math.AP↗

arXiv2013

Equilibrium points of a singular cooperative system with free boundary

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ Δ\u=\frac{\u}{|\u|}χ_{\left\lbrace |\u|>0\right\rbrace}, \qquad \u = (u_1, \cdots, u_m) \ . $$ Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

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arXiv2012

Singularities of axisymmetric free surface flows with gravity

We consider a steady axisymmetric solution of the Euler equations for a fluid (incompressible and with zero vorticity) with a free surface, acted on only by gravity. We analyze stagnation points as well as points on the axis of symmetry. At points on the axis of symmetry which are not stagnation points, constant velocity motion is the only blow-up profile consistent with the invariant scaling of the equation. This suggests the presence of downward pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blow-up profile consistent with the invariant scaling of the equation is Garabedian's pointed bubble solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A fine analysis of the blow-up velocity yields that in the case that the surface is described by an injective curve, the velocity scales almost like $\sqrt{X^2+Y^2+Z^2}$ and is asymptotically given by the velocity field $$V(\sqrt{X^2+Y^2},Z)=c (-\sqrt{X^2+Y^2}, 2Z)$$ with a nonzero constant $c$. The last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by J.-M. Delort. While the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application.

math.AP↗

arXiv2012

Double Obstacle Problems with obstacles given by non-$C^2$ Hamilton-Jacobi equations

We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails and we loose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speed in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi equation $$ \pm |\nabla h-a(x)|^2=\pm 1 \textrm{in} B_1, \qquad h=f \textrm{on} \partial B_1, $$ are in fact $C^{1,α/2}$ provided that $a \in C^α$. This result is optimal and to the authors' best knowledge new.

math.AP↗

arXiv2011

A Remark on the Geometry of Uniformly Rotating Stars

In this paper we classify the free boundary associated to equilibrium configurations of compressible, self-gravitating fluid masses, rotating with constant angular velocity. The equilibrium configurations are all critical points of an associated functional and not necessarily minimizers. Our methods also apply to alternative models in the literature where the angular momentum per unit mass is prescribed. The typical physical model our results apply to is that of uniformly rotating white dwarf stars.

math.AP↗

arXiv2010

On the singularities of a free boundary through Fourier expansion

In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ Δu = - χ_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities - that is points at which the second derivatives of the solution are unbounded - as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity $χ_{\{u>0\}} $. The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ${\mathbb R}^3$. A surprising fact in ${\mathbb R}^3$ is that although $$\frac{u(r\x)}{\sup_{B_1}|u(r\x)|}$$ can converge at singularities to each of the harmonic polynomials $$ xy, {x^2+y^2\over 2}-z^2 \textrm{and} z^2-{x^2+y^2\over 2},$$ it may {\em not} converge to any of the non-axially-symmetric harmonic polynomials $α((1+ δ)x^2 +(1- δ)y^2 - 2z^2)$ with $δ\ne 1/2$. We also prove the existence of stable singularities in ${\mathbb R}^3$.

math.AP↗

arXiv2009

Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of $u$ are unbounded, the level set $\{u=0\}$ consists of two $C^1$-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

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