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Herbert Koch

Publications and source records attributed to Herbert Koch.

At least 37 records · Page 2Linked to original sources
arXiv2018

An asymtotic sharp Sobolev regularity for planar infinity harmonic functions

Given an arbitrary planar $\infty$-harmonic function $u$, for each $α>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^α$, which is sharp as $α\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\infty$-harmonic functions in whole plane.

math.AP↗
arXiv2018

Some sharp Sobolev regularity for inhomogeneous $\infty$-Laplace equation in plane

Suppose $Ω\Subset \mathbb R^2$ and $f\in BV_{loc}(Ω)\cap C^0(Ω)$ with $|f|>0$ in $Ω$. Let $u\in C^0(Ω)$ be a viscosity solution to the inhomogeneous $\infty$-Laplace equation $$ -Δ_{\infty} u :=-\frac12\sum_{i=1}^2(|Du|^2)_iu_i= -\sum_{i,j=1}^2u_iu_ju_{ij} =f \quad {\rm in}\ Ω. $$ The following are proved in this paper. (i) For $ α> 3/2$, we have $|Du|^α\in W^{1,2}_{loc}(Ω)$, which is (asymptotic) sharp when $ α\to 3/2$. Indeed, the function $w(x_1,x_2)=-x_1^ {4/3} $ is a viscosity solution to $-Δ_\infty w=\frac{4^3}{3^4}$ in $\mathbb R^2$. For any $p> 2$, $|Dw|^α\notin W^{1,p}_{loc}(\mathbb R^2)$ whenever $α\in(3/2,3-3/p)$. (ii) For $ α\in(0, 3/2]$ and $p\in[1, 3/(3-α))$, we have $|Du|^α\in W^{1,p}_{loc}(Ω)$, which is sharp when $p\to 3/(3-α)$. Indeed, $ |Dw|^α\notin W^{1,3/(3-α)}_{loc}(\mathbb R^2)$. (iii) For $ ε> 0$, we have $|Du|^{-3+ε}\in L^1_{loc}(Ω)$, which is sharp when $ε\to0$. Indeed, $|Dw|^{-3} \notin L^1_{loc}(\mathbb R^2)$. (iv) For $ α> 0$, we have $$-(|Du|^α)_iu_i= 2α|Du|^{ α-2}f \ \mbox{ almost everywhere in $Ω$}.$$ Some quantative bounds are also given.

math.AP↗
arXiv2018

Renormalization of the two-dimensional stochastic nonlinear wave equations

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a time-dependent renor- malization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW.

math.PR↗
arXiv2018

Conserved energies for the cubic NLS in 1-d

We consider the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each $s>-\frac12$ there exists a conserved energy which is equivalent to the $H^s$ norm of the solution. For the Korteweg-de Vries (KdV) equation there is a similar conserved energy for every $s\ge -1$.

math.AP↗
arXiv2016

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation is the regularity of the solutions and the free boundaries. Here we assume flatness of the solution in space time cylinder and derive smoothness of the interface after a small time, as well as smoothness of the solution in the positivity set and up to the free boundary for some time interval. We use these facts to prove the following eventual regularity result: solutions with compactly supported initial data are smooth after a finite time T that depends on mass and the size of the initial support. This result eliminates the condition of non-degeneracy on the initial data that has been carried on for decades in the literature.

math.AP↗
arXiv2016

Higher Regularity for the fractional thin obstacle problem

In this article we investigate the higher regularity properties of the regular free boundary in the fractional thin obstacle problem. Relying on a Hodograph-Legendre transform, we show that for smooth or analytic obstacles the regular free boundary is smooth or analytic, respectively. This leads to the analysis of a fully nonlinear, degenerate (sub)elliptic operator which we identify as a (fully nonlinear) perturbation of the fractional Baouendi-Grushin Laplacian. Using its intrinsic geometry and adapted function spaces, we invoke the analytic implicit function theorem to deduce analyticity of the regular free boundary.

math.AP↗
arXiv2016

The Variable Coefficient Thin Obstacle Problem: Higher Regularity

In this article we continue our investigation of the thin obstacle problem with variable coefficients which was initiated in \cite{KRS14}, \cite{KRSI}. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove higher order Hölder regularity for the regular free boundary, if the associated coefficients are of the corresponding regularity. For the zero obstacle this yields an improvement of a \emph{full derivative} for the free boundary regularity compared to the regularity of the metric. In the presence of non-zero obstacles or inhomogeneities, we gain \emph{three halves of a derivative} for the free boundary regularity with respect to the regularity of the inhomogeneity. Further we show analyticity of the regular free boundary for analytic metrics. We also discuss the low regularity set-up of $W^{1,p}$ metrics with $p>n+1$ with and without ($L^p$) inhomogeneities. Key new ingredients in our analysis are the introduction of generalized Hölder spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the Baouendi-Grushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) regularity and the splitting technique from \cite{KRSI}.

math.AP↗
arXiv2016

The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary

This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,α}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1}{2}\}}$ regularity of solutions. New ingredients include the use of the Reifenberg flatness of the regular free boundary, the construction of an (almost) optimal barrier function and the introduction of an appropriate splitting of the solution. Important insights depend on the consideration of various intrinsic geometric structures. Based on variations of the arguments in \cite{KRS14} and the present article, we then also discuss the case of non-zero and interior thin obstacles. We obtain the optimal regularity of the solutions and the regularity of the regular free boundary for $W^{1,p}$ metrics and $W^{2,p}$ obstacles with $p\in (2(n+1),\infty]$.

math.AP↗
arXiv2015

Doubling inequalities for the Lamé system with rough coefficients

In this paper we study the local behavior of a solution to the Lamé system when the Lamé coefficients $λ$ and $μ$ satisfy that $μ$ is Lipschitz and $λ$ is essentially bounded in dimension $n\ge 2$. One of the main results is the \emph{local} doubling inequality for the solution of the Lamé system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the \emph{global} doubling inequality, which is useful in some inverse problems.

math.AP↗
arXiv2015

Partial analyticity and nodal sets for nonlinear elliptic systems

We study partial analyticity of solutions to elliptic systems and analyticity of level sets of solutions to nonlinear elliptic systems. We consider several applications, including analyticity of flow lines for bounded stationary solutions to the 2-d Euler equation, and analyticity of water waves with and without surface tension.

math.AP↗
arXiv2015

The Variable Coefficient Thin Obstacle Problem: Carleman Inequalities

In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in the setting of Sobolev metrics, i.e. the coefficients are only $W^{1,p}$ regular for some $p>n+1$. These results provide the basis for our further analysis of the free boundary, the optimal ($C^{1,1/2}$-) regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow-up article in the framework of $W^{1,p}$, $p>2(n+1)$, regular coefficients.

math.AP↗
arXiv2015

Bounded compositions on scaling invariant Besov spaces

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $φ: \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ φ$ is bounded on the homogeneous, scaling invariant Besov space $\dot{B}^s_{n/s,q}(\real^n)$, where the emphasis is on the case $q\not=n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces $\dot{F}^s_{n/s,q}(\real^n)$ with $0 < s < 1$ and $0 < q \leq \infty$.

math.CA↗
arXiv2015

Higher regularity of the free boundary in the elliptic Signorini problem

In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph-Legendre transformation we show that the regular part of the free boundary is real analytic. The first complication in the study is the invertibility of the hodograph transform (which is only $C^{0,1/2}$) which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The second and main complication in the study is that the equation satisfied by the Legendre transform is degenerate. However, the equation has a subelliptic structure and can be viewed as a perturbation of the Baouendi-Grushin operator. By using the $L^p$ theory available for that operator, we can bootstrap the regularity of the Legendre transform up to real analyticity, which imples the real analyticity of the free boundary.

math.AP↗
arXiv2014

Convolution Estimates for Singular Measures and Some Global Nonlinear Brascamp-Lieb Inequalities

We give a $L^2\times L^2 \rightarrow L^2$ convolution estimate for singular measures supported on transversal hypersurfaces in $\mathbb{R}^n$, which improves earlier results of Bejenaru, Herr & Tataru as well as Bejenaru & Herr. The arising quantities are relevant in the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp-Lieb inequalities with explicit constants in the same spirit.

math.CA↗
arXiv2014

Self-similar solutions to super-critical gKdV

We construct self similar finite energy solutions to the slightly super-critical generalized KdV equation. These self similar solutions bifurcate as a function of the exponent $p$ from the soliton at the $L^2$ critical exponent.

math.AP↗
arXiv2014

Semiclassical Lp estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. The revision corrects an exponent in the main theorems.

math-ph↗