arXiv subjects

Daniel Hauer

Publications and source records attributed to Daniel Hauer.

26 records · Page 2Linked to original sources

arXiv2019

Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces

This paper is dedicated to providing new tools and methods for studying the trend to equilibrium of gradient flows in metric spaces in the entropy and metric sense, to establish decay rates, finite time of extinction, and to characterize Lyapunov stable equilibrium points. In addition we outline that the celebrated Entropy-Entropy production inequality used in kinetic theory is nothing less than a global Kurdyka-Lojasiewicz-Simon inequality. This links two different areas, namely, algebraic geometry with kinetic theory. As an application of the tools developed in this paper, we obtain the following results: - New upper bounds on the extinction time of gradient flows associated with the total variational flow. - If the metric space is the p-Wasserstein space, then new HWI-, Talagrand-, and logarithmic Sobolev inequalities are obtained for functionals associated with nonlinear diffusion problems modeling drift, potential and interaction phenomena. - It is shown that these inequalities are equivalent to the Kurdyka-Lojasiewicz-Simon inequality and hence, they imply trend to equilibrium of the gradient flows with decay rates or arrival in finite time.

math.AP↗

arXiv2019

Regularizing effect of homogeneous evolution equations: case homogeneous order zero

In this paper, we develop a functional analytical theory for establishing that mild solutions of first-order Cauchy problems involving homogeneous operators of order zero are strong solutions; in particular, the first-order time derivative satisfies a global regularity estimate depending only on the initial value and the positive time. We apply those results to the Cauchy problem associated with the total variational flow operator and the nonlocal fractional 1-Laplace operator.

math.AP↗

arXiv2018

Fractional powers of monotone operators in Hilbert spaces

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $\textrm{Rg}(A)$ of $A$, then for every $0<s<1$, the Dirichlet problem associated with the Bessel-type equation $$ A_{1-2s}u:=-\frac{1-2s}{t}u_{t}-u_{tt}+Au\ni 0 $$ is well-posed for boundary values $φ\in \overline{D(A)}^{\mbox{}_{H}}$. This allows us to define the Dirichlet-to-Neumann (DtN) operator $Λ_{s}$ associated with $A_{1-2s}$ as $$ φ\mapsto Λ_{s}φ:=-\lim_{t\to 0+}t^{1-2s}u_{t}(t)\qquad\text{in H.} $$ The existence of the DtN operator $Λ_{s}$ associated with $A_{1-2s}$ is the first step to define fractional powers $A^α$ of monotone (possibly, nonlinear and multivalued) operators $A$ on $H$. We prove that $Λ_{s}$ is monotone on $H$ and if $\overlineΛ_{s}$ is the closure of $Λ_{s}$ in $H\times H_{w}$ then we provide sufficient conditions implying that $-\overlineΛ_{s}$ generates a strongly continuous semigroup on $\overline{D(A)}^{\mbox{}_{H}}$. In addition, we show that if $A$ is completely accretive on $L^{2}(Σ,μ)$ for a $σ$-finite measure space $(Σ,μ)$, then $Λ_{s}$ inherits this property from $A$.

math.AP↗

arXiv2018

Non-concavity of Robin eigenfunctions

On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground stat is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.

math.AP↗

arXiv2016

Regularisation effects of nonlinear semigroups

One introduces natural and simple methods to deduce $L^{s}$-$L^{\infty}$-re\-gularisation estimates for $1\le s< \infty$ of nonlinear semigroups holding uniformly for all time with sharp exponents from natural Gagliardo-Nirenberg inequalities. From $L^{q}$-$L^{r}$ Gagliardo-Nirenberg inequalities, $1\le q, r\le \infty$, one deduces $L^{q}$-$L^{r}$ estimates for the semigroup. New nonlinear interpolation techniques of independent interest are introduced in order to extrapolate such estimates to $L^{\tilde{q}}$-$L^{\infty}$ estimates for some $\tilde{q}$, $1\le \tilde{q}<\infty$. Finally one is able to extrapolate to $L^{s}$-$L^{\infty}$ estimates for $1\le s<q$. The theory developed in this monograph allows to work with minimal regularity assumptions on solutions of nonlinear parabolic boundary value problems as illustrated in a plethora of examples including nonlocal diffusion processes.

math.AP↗

arXiv2015

Nonlinear semigroups generated by $j$-elliptic functionals

We generalise the theory of energy functionals used in the study of gradient systems to the case where the domain of definition of the functional cannot be embedded into the Hilbert space $H$ on which the associated operator acts, such as when $H$ is a trace space. We show that under weak conditions on the functional $φ$ and the map $j$ from the effective domain of $φ$ to $H$, which in opposition to the classical theory does not have to be injective or even continuous, the operator on $H$ naturally associated with the pair $(φ,j)$ nevertheless generates a nonlinear semigroup of contractions on $H$. We show that this operator, which we call the $j$-subgradient of $φ$, is the (classical) subgradient of another functional on $H$, and give an extensive characterisation of this functional in terms of $φ$ and $j$. In the case where $H$ is an $L^2$-space, we also characterise the positivity, $L^\infty$-contractivity and existence of order-preserving extrapolations to $L^q$ of the semigroup in terms of $φ$ and $j$. This theory is illustrated through numerous examples, including the $p$-Dirichlet-to-Neumann operator, general Robin-type parabolic boundary value problems for the $p$-Laplacian on very rough domains, and certain coupled parabolic-elliptic systems.

math.FA↗

arXiv2014

A Liouville theorem for $p$-harmonic functions on exterior domains

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace equation.

math.AP↗

arXiv2012

New weighted Hardy's inequalities with application to non-existence of global solutions

In this article, we prove a weighted Hardy inequality for $1 d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.

math.AP↗