arXiv subjects

Tim Böhnlein

Publications and source records attributed to Tim Böhnlein.

5 recordsLinked to original sources

arXiv2025

Meyers exponent rules the first-order approach to second-order elliptic boundary value problems

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy-Riemann system in complex analysis. It hinges on global Lp -bounds for some p > 2 for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some p > 2, expressed in terms of weak reverse H{ö}lder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

math.AP↗

arXiv2024

Gaussian estimates vs. elliptic regularity on open sets

Given an elliptic operator $L= - \mathrm{div} (A \nabla \cdot)$ subject to mixed boundary conditions on an open subset of $\mathbb{R}^d$, we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of $L$-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet we prove consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

math.AP↗

arXiv2024

Bounded functional calculus for divergence form operators with dynamical boundary conditions

We consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a matter of fact, the elliptic operator and its semigroup act simultaneously in the interior and on the boundary. We show that the elliptic operator has a bounded $\mathrm{H}^\infty$-calculus in $\mathrm{L}^p$ if the coefficients satisfy a $p$-adapted ellipticity condition. A major challenge in the proof is that different parts of the spatial domain of the operator have different dimensions. Our strategy relies on extending a contractivity criterion due to Nittka and a non-linear heat flow method recently popularized by Carbonaro-Dragičević to our setting.

math.AP↗

arXiv2023

Hardy spaces adapted to elliptic operators on open sets

Let $L= - \mathrm{div} (A \nabla \cdot)$ be an elliptic operator defined on an open subset of $\mathbb{R}^d$, complemented with mixed boundary conditions. Under suitable assumptions on the operator and the geometry, we derive an atomic characterization (depending only on the boundary conditions) for the Hardy space $H^1_L$ defined using an adapted square function for $L$. This generalizes known results of Auscher and Russ in the case of pure Dirichlet/Neumann boundary conditions on Lipschitz domains. In particular, we develop a connection between the harmonic analysis of $L$ and its underlying geometry.

math.FA↗

arXiv2023

Explicit improvements for $\mathrm{L}^p$-estimates related to elliptic systems

We give a simple argument to obtain $\mathrm{L}^p$-boundedness for heat semigroups associated to uniformly strongly elliptic systems on $\mathbb{R}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give $p$ explicitly in terms of ellipticity. It is optimal at the endpoint $p=\infty$. We also obtain $\mathrm{L}^p$-estimates for the gradient of the semigroup, where $p>2$ depends on ellipticity but not on dimension.

math.AP↗