arXiv subjects

Daniel Hauer

Publications and source records attributed to Daniel Hauer.

At least 19 recordsLinked to original sources

arXiv2026

Fundamental Gaps for the Dirichlet p-Laplacian with Convex Potentials: Sharp One-Dimensional Bounds and a Higher-Dimensional Dichotomy

We study fundamental gaps for the Dirichlet $p$-Laplacian on bounded convex domains with convex potentials. In one dimension, we prove the sharp inequality \[ \lambda_{2,p}(I_D,V)-\lambda_{1,p}(I_D,V) \geq (p-1)(2^p-1)\left(\frac{\pi_p}{D}\right)^p \] for every $p>1$ and every convex potential, with equality precisely for constant potentials. For $N\geq2$, we identify a sharp transition at $p=2$ through collapsing smooth convex domains: the gap vanishes for $1 2$. In the regime $p\geq2$, we prove log-concavity of the positive first eigenfunction by a regularization and two-point maximum principle. We then establish a degenerate weighted Poincar\'e inequality, which yields quantitative stability estimates for the $L^p$-Poincar\'e inequality and, in turn, quantitative lower bounds for the fundamental gap. For zero potential, we further obtain an enhanced gap estimate involving both the first eigenvalue and the diameter. Finally, we prove existence of diameter-normalized gap minimizers for $p>2$ and show that they degenerate as $p\downarrow2$, whereas for $p=2$ the optimal gap is not attained by any bounded $N$-dimensional convex domain.

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arXiv2026

Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian

We investigate nonnegative mild solutions of $\partial_t u+(-\Delta)^{\ln}u=f(u)$ in $(0,T)\times\mathbb R^N$, with initial datum $u(0,\cdot)=\mu u_0$, $\mu>0$. Unlike the classical and fractional heat semigroups, the positive logarithmic heat kernel exists only for $0<t<N/2$, and the corresponding linear evolution may become singular at its terminal time, with lifespan and growth depending on the spatial decay of $u_0$. The behavior of $f$ near zero determines local solvability: if $\int_{0^+} d\sigma/f(\sigma)<\infty$, then no finite nonnegative solution exists on any positive time interval. Under suitable assumptions on $u_0$ and $f$, we establish well-posedness for the nonintegrable logarithmic heat kernel. If $f$ has at most global linear growth, the nonlinear solution attains the full linear lifespan, while the Osgood condition at infinity implies that the maximal existence time tends to zero as $\mu\to\infty$. We further distinguish slow-decay, fast-decay, and critical-tail initial data. In the noncritical regimes, a weighted Osgood tail condition yields blow-up strictly before the linear terminal time; if it fails, an amplitude threshold occurs under additional assumptions on $f$. In the critical regime, the dividing power is $3/2$: a square-root weighted Osgood condition yields premature blow-up, while its failure again leads to an amplitude threshold. Finally, we obtain terminal-time blow-up estimates and sharp rates for power nonlinearities.

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arXiv2026

The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities

In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter $s\in(0,1)$. We investigate its fundamental analytic and spectral properties, including its relation to the conformal logarithmic Laplacian, its spectral representation, and the explicit form of its eigenvalues and eigenfunctions. We further establish its conformal covariance law and derive the associated Yamabe-type equation, proving its equivalence to the corresponding conformal equation in $\mathbb R^N$ through stereographic projection. Finally, we apply this framework to sharp Sobolev-type inequalities, recovering the sharp logarithmic Sobolev inequality, revealing the failure of a naive fractional--logarithmic analogue, and establishing new sharp fractional--logarithmic inequalities.

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arXiv2026

The Fractional-Logarithmic Laplacian:Fundamental Properties and Eigenvalues

In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian $ (-\Delta)^{s+\log} $, defined as the derivative of the fractional Laplacian $ (-\Delta)^t $ at $ t=s $. It is a singular integral operator with Fourier symbol $ |\xi|^{2s}(2\ln|\xi|) $, and we prove the pointwise integral representation \[ (-\Delta)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-\Delta)^s u(x), \] where $ c_{n,s} $ is the normalization constant of the fractional Laplacian and $ b_{n,s}:=\frac{d}{ds}c_{n,s}.$ We also establish several equivalent formulations of $ (-\Delta)^{s+\log} $, including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization. We develop the associated functional framework on both $ \mathbb{R}^n $ and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent $ 2_s^*=\frac{2n}{n-2s},$ a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and $ L^\infty $-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the $ k $-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian.

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arXiv2025

A multi-point maximum principle to prove global Harnack inequalities for Schr\"odinger operators

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the linear Schr\"odinger equation on a Riemannian manifold $M$ with nonnegative Ricci curvature, where the potential term $V$ is bounded from below. Our approach is based on a multi-point maximum principle argument. Standard proofs of this result (see, for instance, Li-Yau [Acta Math, 1986]) rely on first establishing a gradient estimate. This requires the solution to be at least $C^4$ on $M$. We instead prove the Harnack inequality directly, which has the advantage of avoiding higher-order derivatives of the solution in the proof, enabling us to assume it is only $C^2$ on $M$. In the particular case that $V$ is the quadratic potential $V(x)=|x|^2$ and $M$ is the Euclidean space $\mathbb{R}^d$, we prove a new Harnack inequality with sharper constants. Finally, we treat positive solutions of the Schr\"odinger equation with a gradient drift term, including applications to the Ornstein-Uhlenbeck operator $\Delta - x\cdot \nabla$ with quadratic potential in $\mathbb{R}^d$.

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arXiv2024

On some Liouville theorems for p-Laplace type operators

The goal of this note is to consider Liouville type theorem for p-Laplacian type operators. In particular guided by the Laplacian case one establishes analogous results for the p-Laplacian and operators of this type.

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arXiv2024

Existence of an extremal function of Sobolev critical embedding with an $\alpha$-homogeneous weight

In our previous publication [{\em Calc. Var. Partial Differential Equations}, 60(1):Paper No. 16, 27, 2021], we delved into examining a critical Sobolev-type embedding of a Sobolev weighted space into an exponential weighted Orlicz space. We specifically determined the optimal Moser-type constant for this embedding, utilizing the monomial weight introduced by Cabr\'e and Ros-Oton [{\em J. Differential Equations}, 255(11):4312--4336, 2013]. Towards the conclusion of that paper, we pledged to explore the existence of an extremal function within this framework. In this current work, we not only provide a positive affirmation to this inquiry but extend it to a broader range of weights known as \emph{$\alpha$-homogeneous weights}.

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arXiv2024

Regularity and Separation for $p$-Laplace operators

We analyze $p$-Laplace operators with degenerate elliptic coefficients. This investigation includes Gru\v{s}in type $p$-Laplace operators. We describe a \emph{separation phenomenon} in elliptic and parabolic $p$-Laplace type equations, which provides an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for considered setting does not imply continuity of elliptic equations. On the other hand, we are able to establish global $L^1$-$L^\infty$-regularization and decay estimates of every mild solution of the parabolic Gru\v{s}in type $p$-Laplace equation.

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arXiv2023

An extension problem for the logarithmic Laplacian

The logarithmic Laplacian on the (whole) N-dimensional Euclidean space is defined as the first variation of the fractional Laplacian of order 2s at s=0 or, alternatively, as a singular Fourier integral operator with logarithmic symbol. While this operator has attracted fastly growing attention in recent years due to its relevance in the study of order-dependent problems, a characterization via a local extension problem on the (N+1)-dimensional upper half-space in the spirit of the Cafferelli-Sivestre extension for the fractional Laplacian has been missing so far. In this paper, we establish such a characterization. More precisely, we show that, up to a multiplicative constant, the logarithmic Laplacian coincides with the boundary-value operator associated with a weighted second-order operator on the upper half-space, which maps inhomogeneous Neumann data to a Robin boundary-value of the corresponding distributional solution with a singular excess term. This extension property of the logarithmic Laplacian leads to a new energy functional associated with this operator. By doubling the extension-variable, we show that distributional solutions of the extension problem are actually harmonic in the (N+2)-dimensional Euclidean space away from the boundary. As an application of these results, we establish a weak unique continuation principle for the (stationary) logarithmic Laplace equation.

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arXiv2023

Characterization of solutions of a generalized Helmholtz problem

In this article, we classify all distributional solutions of $f(-\Delta)u=f(1)u$ where $f$ is a non-constant Bernstein function. Specifically, we show that the Fourier transform of $u$ is a single-layer distribution on the unit sphere. Examples of such operators include $(-\Delta)^\sigma$ (for $\sigma \in (0,1]$), $\log(1-\Delta)$ and $(-\Delta)^\frac{1}{2}\text{tanh}((-\Delta)^\frac{1}{2})$.

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arXiv2023

The Barenblatt solution of an evolution problem governed by a doubly nonlinear nonlocal operator

In this article, we prove existence and uniqueness of the Barenblatt solution of the evolution equation on the whole Euclidean space where the principle part is the nonlocal fractional p-Laplacian composed with a power function. Our proof generalizes methods developped by J.-L. Vazquez [Nonlinear Anal., 199 (2022), Calc. Var. Partial Differential Equations, 60 (2021)] for the evolution equation driven by the fractional p-Laplacian on the whole Euclidean space. In particular, we required an Aleksandrov symmetry principle, which can be applied to the mild solutions of the evolution equation in $L^1$ governed by the doubly nonlinear nonlocal operator, and the construction of global barrier functions. The Aleksandrov symmetry principle might be of independent interest.

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arXiv2023

Density of the domain of doubly nonlinear operators in $L^1$

The aim of this paper is to provide sufficient conditions implying that the effective domain $D(A\phi)$ of an $m$-accretive operator $A\phi$ in $L^1$ is dense in $L^1$. Here, $A\phi$ refers to the composition $A\circ \phi$ in $L^1$ of the part $A=(\partial\mathcal{E})_{\vert L^{1\cap \infty}}$ in $L^{1\cap\infty}\times L^{1\cap\infty}$ of the subgradient $\partial\mathcal{E}$ in $L^2$ of a convex, proper, lower semicontinuous functional $\mathcal{E}$ on $L^2$ and a continuous, strictly increasing function $\phi$ on the real line $\mathbb{R}$. To illustrate the role of the sufficient conditions, we apply our main result to the class of doubly nonlinear operators $A\phi$, where $A$ is a classical Leray-Lions operator.

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arXiv2021

A doubly nonlinear evolution problem involving the fractional p-Laplacian

In this article, we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional $p$-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in $\mathbb{R}^{d}$ and composed with a continuous, strictly increasing function. We establish well-posedness in $L^1$ in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain $L^{q}$-$L^{\infty}$ regularity estimates for mild solutions, implying decay estimates and extending the property of strong solutions for more initial data. Moreover, we prove local and global H\"older continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.

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arXiv2021

Functional Calculus via the extension technique: a first hitting time approach

In this article, we present a solution to the problem: "Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension problem?", which was raised in the pioneering work [Comm. Par.Diff. Equ. 32 (2007)] by Caffarelli and Silvestre. In fact, we even go a step further by replacing the negative Laplace operator $-\Delta$ on $\mathbb{R}^{d}$ by an $m$-accretive operator $A$ on a general Banach space $X$ and the Dirichlet-to-Neumann operator by the Dirichlet-to-Wentzell operator. We establish uniqueness of solutions to the extension problem in this general framework, which seems to be new in the literature and independent interest. The aim of this paper is to provide a new Phillips-Bochner type functional calculus which uses probabilistic tools from excursion theory. With our method, we are able to characterize all linear operators $\psi(A)$ among the class $CBF$ of complete Bernstein functions $\psi$, resulting in a new characterization of the famous Phillips' subordination theorem within this class $CBF$.

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arXiv2020

Regularizing effect of homogeneous evolution equations with perturbation

Since the pioneering works by Aronson & B\'enilan [C. R. Acad. Sci. Paris S\'er., 1979] and B\'enilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous at every positive time. Moreover, if the underlying Banach space has the Radon-Nikod\'ym property, then these mild solution is a.e. differentiable, and the time-derivative satisfies global and point-wise bounds. In this paper, we show that these results remain true if the homogeneous operator is perturbed by a Lipschitz continuous mapping. More precisely, we establish global $L^1$ Aronson-B\'enilan type estimates and point-wise Aronson-B\'enilan type estimates. We apply our theory to derive global $L^q$-$L^{\infty}$-estimates on the time-derivative of the perturbed diffusion problem governed by the Dirichlet-to-Neumann operator associated with the $p$-Laplace-Beltrami operator and lower-order terms on a compact Riemannian manifold with a Lipschitz boundary.

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arXiv2020

The fundamental gap for a one-dimensional Schr\"odinger operator with Robin boundary conditions

For Schr\"odinger operators on an interval with either convex or symmetric single-well potentials, and Robin or Neumann boundary conditions, the gap between the two lowest eigenvalues is minimised when the potential is constant. We also have results for the $p$-Laplacian.

math.CA↗

arXiv2019

The Dirichlet-to-Neumann operator associated with the $1$-Laplace operator and evolution problems

We present first results on the Dirichlet-to-Neumann operator associated with the $1$-Laplace operator in $L^1$. In particular, we show that this operator can be realized as a sub-differential operator in $L^1\times L^{\infty}$ of a homogeneous convex, continuous functional with effective domain $L^1$. Even though the Dirichlet problem associated with the $1$-Laplace operator loses the property that weak solutions for boundary data in $L^1$ are unique, we prove a type of stability/compactness result with respect to the boundary data in $L^1$ of this problem. We apply our results for the stationary Dirichlet problem to evolution problems governed by the Dirichlet-to-Neumann operator, which can equivalently be formulated as singular coupled elliptic-parabolic initial boundary-value problems. For initial data in $L^q$, $1\le q\le \infty$, we obtain well-posedness, that every mild solution is, indeed, a strong solution, and establish long-time stability of the semigroup generated by the negative Dirichlet-to-Neumann operator associated with the $1$-Laplace operator.

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arXiv2019

Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\varphi$ has a smoothing effect, discovered by H. Br\'ezis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $\varphi$ are not only closed but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

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