arXiv subjects
Publications and source records attributed to Simon Nowak.
Solutions to nonlinear nonlocal systems of order $2s>1$ in $\mathbb{R}^n$ are $C^{1,\alpha}$, for every $\alpha <2s-1$, outside a closed singular set whose Hausdorff dimension is less than $n-2$, and which is empty when $n=2$.
In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivi\`ere and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.
In this work, we provide a comprehensive gradient regularity theory for a broad class of nonlinear kinetic Fokker-Planck equations. We achieve this by establishing precise pointwise estimates in terms of the data in the spirit of nonlinear potential theory, leading to fine gradient regularity results under borderline assumptions on the data. Notably, our gradient estimates are novel already in the absence of forcing terms and even for linear kinetic Fokker-Planck equations in divergence form.
Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allows to derive fine properties of solutions under sharp assumptions on data and kernel coefficients.
The primary objective of this work is to establish pointwise gradient estimates for solutions to a class of parabolic nonlinear nonlocal measure data problems, expressed in terms of caloric Riesz potentials of the data. As a consequence of our pointwise estimates, we obtain that the first-order regularity properties of solutions to such general parabolic nonlinear nonlocal equations, both in terms of size and oscillations of the spatial gradient, closely resemble the ones of the fractional heat equation even at highly refined scales. Along the way, we show that solutions to homogeneous parabolic nonlinear nonlocal equations have H\"older continuous spatial gradients under optimal assumptions on the nonlocal tails.
In this work, we study the higher differentiability of solutions to the inhomogeneous fractional $p$-Laplace equation under different regularity assumptions on the data. In the superquadratic case, we extend and sharpen several previous results, while in the subquadratic regime our results constitute completely novel developments even in the homogeneous case. In particular, in the local limit our results are consistent with well-known higher differentiability results for the standard inhomogeneous $p$-Laplace equation. All of our main results remain valid in the vectorial context of fractional $p$-Laplace systems.
We extend the De Giorgi iteration technique to the vectorial setting. For this we replace the usual scalar truncation operator by a vectorial shortening operator. As an application, we prove local boundedness for local and nonlocal nonlinear systems. Furthermore, we show convex hull properties, which are a generalization of the maximum principle to the case of systems.
The aim of this work is to establish numerous interrelated gradient estimates in the nonlinear nonlocal setting. First of all, we prove that weak solutions to a class of homogeneous nonlinear nonlocal equations of possibly arbitrarily low order have H\"{o}lder continuous gradients. Using these estimates in the homogeneous case, we then prove sharp higher differentiability as well as pointwise gradient potential estimates for nonlinear nonlocal equations of order larger than one in the presence of general measure data. Our pointwise estimates imply that the first-order regularity properties of such nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian.
The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow to derive new estimates for the dissipative surface quasi-geostrophic equation.
We prove fine higher regularity results of Calder\'on-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
We consider nonlocal equations of order larger than one with measure data and prove gradient regularity in Sobolev and H\"older spaces as well as pointwise bounds of the gradient in terms of Riesz potentials, leading to fine regularity results in many commonly used function spaces. The kernel of the integral operators involves a H\"older dependence in the variables and is not assumed to be translation invariant.
This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In this paper, we follow this direction and show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions $u \in W^{s,2}$ in fact belong to $W^{t,p}_{loc}$ for any $s \leq t < \min\{2s,1\}$, where $p>2$ reflects the amount of integrability gained. In other words, our gain of differentiability does not depend on the amount of integrability we are able to gain. This extends numerous results in previous works, where either continuity of the coefficient was required or only an in general smaller gain of differentiability was proved.
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher H\"older regularity for such nonlocal equations.
We prove a higher regularity result for weak solutions to nonlinear nonlocal equations along the integrability scale of Bessel potential spaces $H^{s,p}$ under a mild continuity assumption on the kernel. By embedding, this also yields regularity in Sobolev-Slobodeckij spaces $W^{s,p}$. Our approach is based on a characterization of Bessel potential spaces in terms of a certain nonlocal gradient-type operator and a perturbation approach commonly used in the context of local elliptic equations in divergence form.
We study the higher H\"older regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as follows. First, we use certain test functions that involve discrete fractional derivatives in order to obtain higher H\"older regularity for homogeneous equations driven by a locally translation invariant kernel, while the global behaviour of the kernel is allowed to be more general. This enables us to deduce the desired regularity in the general case by an approximation argument.
In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces $H^{s,p}$. Our main results can be seen as an extension of the well-known $W^{1,p}$ regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.