Special Session 108: Regularity in local versus nonlocal problems

Interior Gradient Estimates for Nonhomogeneous Parabolic p-Laplace Systems

Pedra Andrade Paris Lodron University of Salzburg Austria Co-Author(s):    Verena Boegelein, Frank Duzaar and Kristian Moring

Abstract: In this talk, I will discuss gradient estimates for weak solutions to nonhomogeneous parabolic $p$-Laplacian systems with a right-hand side in non-divergence form. Under suitable regularity conditions on the coefficients, we obtain improved integrability properties beyond those currently known in the literature. Our approach combines classical techniques, including comparison estimates and stopping-time arguments, to derive Calderon-Zygmund-type estimates.

Schauder estimates for parabolic $p$-Laplace systems

Verena Bogelein University of Salzburg Austria Co-Author(s):    Frank Duzaar, Ugo Gianazza, Naian Liao, Christoph Scheven

Abstract: In this talk, we consider parabolic systems of $p$-Laplace type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(\mu^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $\mu\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and H\older continuous with respect to the spatial variable $x$. Via Schauder estimates we establish local H\older continuity of the spatial gradient of bounded weak solutions. As an application, we derive H\older estimates for the gradient of weak solutions to a doubly nonlinear parabolic equation in the supercritical fast diffusion regime. In particular, we obtain quantitative bounds for the spatial gradient and its H\older continuity. This is joint work with F.~Duzaar, U.~Gianazza, N.~Liao, and C.~Scheven.

Holder regularity for a class of doubly non linear PDEs

Filippo Maria Cassanello Universita degli studi di Cagliari Italy Co-Author(s):    Filippo Maria Cassanello

Abstract: We prove local Holder continuity for non negative, locally bounded, local weak solutions for the class of doubly nonlinear parabolic equations $\partial_t(u^q) + \text{Div}(|Du|^{p-2}Du) =0$ with $p>2$ and $p-1>q>0$. The novelty of the proof of such result is given by the use of an expansion of positivity argument, combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated to the problem.

Manifold-valued minimizers of double phase functionals: partial regularity and Lavrentiev gap

Filomena De Filippis University of Salzburg Austria Co-Author(s):

Abstract: Two classical research topics in the calculus of variations are the study of regularity results for minimizers, with values in a manifold, of functionals with $p$-growth, whose foundations date back to the works of Eells \& Sampson, Schoen \& Uhlenbeck, Fuchs, Hardt \& Lin and Luckhaus, and the density of smooth maps between compact manifolds in Sobolev spaces, characterized by Bethuel, Hang \& Lin, and Hajlasz in terms of the topological properties of the target. In this talk we extend these themes to the nonuniformly elliptic setting of functionals with $(p,q)$-growth, with particular emphasis on double phase energies. We show that vector-valued minimizers subject to certain manifold constraints enjoy partial regularity of the gradient, that is, regularity of the solutions outside a negligible closed subset. We further investigate necessary and sufficient conditions for the density of smooth maps between suitable compact manifolds in nonhomogeneous spaces characterized by the finiteness of anisotropic energies such as those of double phase type. From joint works with C. A. Antonini, A. Nastasi, and C. Pacchiano Camacho.

Non-Diagonal Quasilinear Degenerate Elliptic Systems: Existence and Regularity

Patrizia Di Gironimo Dipartimento di Matematica, Università di Salerno Italy Co-Author(s):

Abstract: In this talk, we discuss non-diagonal vectorial quasilinear degenerate elliptic systems defined on a bounded open set $\Omega \subset mathbb{R}^n$, with $n > 2$, of the form \begin{equation} \label{eq_divergenza} \begin{cases} - \text{div} (a(x,u(x))Du(x))= f(x) ,& x \in \Omega \ u=0, & x \in \partial \Omega \end{cases} \end{equation} where $u, f : \Omega \to \mathbb{R}^N$ and $a: \Omega \times \mathbb{R}^N \to \mathbb{R}^{N^2 n^2}$ is a bounded Carath\`{e}odory function. Unlike the scalar case, in the vectorial setting $(N \ge 2)$ one cannot generally expect boundedness or high integrability of solutions, even for regular data. We will show how, by imposing conditions on the support of the off-diagonal coefficients, it is possible to guarantee existence of solutions whose regularity depends on the datum f. In particular, we will analyze the case where f belongs to an appropriate Marcinkiewicz space, highlighting regularity properties for the systems under consideration.

Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients

Frank Duzaar University of Salzburg Austria Co-Author(s):    Verena Boegelein, Ugo Gioanazza, Naian Liao

Abstract: A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,\sfA(x,t,Du)=0 \quad\text{in }E_T\subset \R^N\times\R, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1\frac{N(2-p)}{p}$, we derive sharp, scale-invariant $L^\infty$-estimates. \emph{Higher integrability of the gradient:} $|Du|$ self-improves from $L^p_{\rm loc}$ to $L^{p(1+\varepsilon)}_{\mathrm{loc}}$ for some $\varepsilon>0$ depending only on the data. The same results still hold given proper source terms.

Regularity Results for Weak Solutions of Singular Parabolic-Elliptic Chemotaxis Models with a Source Term

Fatma Gamze DUZGUN Cagliari University Italy Co-Author(s):

Abstract: We study a quasilinear parabolic-elliptic chemotaxis system featuring singular porous medium-type diffusion, a nonlinear chemotactic sensitivity, and a source term that depends on both the cell density and its gradient. By applying the De Giorgi-DiBenedetto method, we demonstrate that the bounded solutions exhibit Holder continuity.

Towards local regularity for signed solutions to doubly nonlinear parabolic equations involving nonlocal operators

Simona Fornaro University of Pavia Italy Co-Author(s):    Eurica Henriques

Abstract: In this talk I will present some preliminary results in order to study the local continuity of signed weak solutions to a class of parabolic doubly non linear equations involving a non local operator of $p$-Laplacian type $$ \partial_t(|u|^{q-1}u) +L u=0 $$ where $q>0$ and $$ L u(x,t)={\rm P.V.} \int_{\mathbb{R}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))dy, $$ with $p>1$, $K:\mathbb{R}^N\times\mathbb{R}^N\times (0,T]\to [0,+\infty)$ measurable kernel, such that $$ \frac{C_0}{|x-y|^{N+sp}}\le K(x,y,t)=K(y,x,t)\le \frac{C_1}{|x-y|^{N+sp}}\quad \text{for a.a.} x,y\in \mathbb{R}^N $$ for some positive constants $C_0,C_1$ and $s\in (0,1)$. The final aim consists in proving the H\older estimate under a minimal tail condition, in the same spirit as in [1]. The approach is structural and relies on the so-called intrinsic geometry. Moreover it is quite general and we believe that it is possible to obtain a general modulus of continuity. [1]{N. Liao, }{\it On the modulus of continuity of solutions to nonlocal parabolic equations,} {J. Lond. Math. Soc. (2)},{110 (3)}{(2024), Paper No. e12985, 30 pp}.

On the H\older differentiability of fractional $p$-harmonic functions

Davide Giovagnoli University of Bologna Italy Co-Author(s):    David Jesus and Luis Silvestre

Abstract: In this talk, we present a recent result, obtained with David Jesus (KAUST) and Luis Silvestre (University of Chicago), establishing that solutions to the homogeneous fractional p-laplace equation enjoy H\older differentiability whenever 2 \leq p < \frac{2}{1-s}, extending the well-known regularity theory from the local to the nonlocal setting.

Different assumptions in vectorial elliptic problems with a focus on the quasilinear case

Marta Macr\`i University of L`Aquila Italy Co-Author(s):

Abstract: This talk presents several structural assumptions used to prove existence and regularity for vectorial elliptic systems. In nonlinear settings, Uhlenbeck-type conditions, Landes-type assumptions, and more recent componentwise coercivity are frequently employed. In the quasilinear case, however, such hypotheses force all off diagonal coefficients to vanish, effectively decoupling the system. To address genuinely quasilinear, non-diagonal systems, one often imposes conditions on the support of the off diagonal coefficients. As a specific example, an existence and regularity result will be discussed for a quasilinear elliptic system with a drift term.

Unified regularity properties for minimizers under double-phase and exponential growth

Antonella Nastasi University of Palermo Italy Co-Author(s):    Paolo Marcellini (University of Florence) and Cintia Pacchiano Camacho (Universidad Nacional Autonoma de Mexico)

Abstract: In this talk we will present some general growth conditions for functionals, including the so-called natural growth, or polynomial, or growth conditions, or even exponential growth, in order to obtain that any local minimizer of the corresponding energy integral is locally Lipschitz continuous. In fact this is the fundamental step for further regularity; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of non-uniform elliptic variational problems to a context of uniform ellipticity. This is a joint work with Paolo Marcellini and Cintia Pacchiano Camacho.

Carleson-type removability for $p$-parabolic equations

Leah Schaetzler Aalto University Finland Co-Author(s):    Micha{\l} Borowski, Theo Elenius, and David Stolnicki

Abstract: In this talk, I will characterize removable sets for H{\o}lder continuous solutions to degenerate PDEs of parabolic $p$-Laplace type. I will give sufficient and necessary conditions for a set to be removable in terms of an intrinsic parabolic Hausdorff measure, which depends on the considered H{\o}lder exponent. Our method of proof for the sufficient condition only relies on fundamental properties of the obstacle problem and supersolutions and is applicable to a wide class of operators. To obtain the necessary condition, we establish the H{\o}lder continuity of solutions to measure data problems under a certain decay assumption on the considered measure. The talk is based on joint work with Micha{\l} Borowski, Theo Elenius, and David Stolnicki.

Fractional diffusion in heterogeneous media with a discontinuous coefficient

Ana Jacinta Soares Centre of Mathematics, University of Minho Portugal Co-Author(s):    E. Abreu, A. Azevedo, J. Guevara, L. Santos

Abstract: We consider a flow problem in heterogeneous porous media with strongly varying properties. Such situations naturally arise in practical applications, particularly in petroleum reservoir engineering and groundwater flow, where abrupt changes in the medium induce discontinuous coefficients. This motivates the study of a nonlocal diffusion model involving a fractional gradient operator with a spatially varying and discontinuous coefficient $\beta(x) \geq \beta_* > 0$, capturing multiscale effects and anomalous transport phenomena. We construct a family of approximating problems by combining a regularizing diffusion term with a suitable approximation procedure. The analysis is carried out in an appropriate functional framework for weak solutions. We establish existence, positivity, and continuous dependence on the data, and derive uniform a priori estimates. These estimates allow us to pass to the limit in the approximating problems and obtain a solution to the limiting problem. This is joint work with Eduardo Abreu, Assis Azevedo, Julio Guevara, and Lisa Santos.

Improved moduli of continuity for degenerate phase transitions

Jose Miguel Urbano KAUST Saudi Arabia Co-Author(s):    Ugo Gianazza (Pavia) and Naian Liao (Salzburg)

Abstract: We improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the $N-$dimensional, two-phase Stefan problem featuring a $p-$degenerate diffusion: for $p=N\geq 3$, we sharpen it to $$\boldsymbol{\omega}(r) \approx \exp (-c| \ln r|^{\frac1N});$$ for $p>\max\{2,N\}$, we derive an unexpected H\"older modulus.

Existence and uniqueness of weak solutions to singular anisotropic elliptic problems

Eugenio Vecchi ALMA MATER STUDIORUM - Universita' di Bologna Italy Co-Author(s):

Abstract: We consider a quasilinear singular anisotropic elliptic problem. Motivated by the celebrated paper by Lazer and McKenna, we provide necessary and sufficient conditions for the existence of weak solutions. We also discuss uniqueness of weak solutions. The talk is based on a joint work with F. Esposito and F. Oliva.

Fundamental regularity results for widely degenerate, doubly nonlinear anisotropic diffusion equations

Matias Vestberg Uppsala Universitet Sweden Co-Author(s):    Pasquale Ambrosio, Simone Ciani

Abstract: We investigate fundamental regularity properties of solutions to a class of widely degenerate, doubly nonlinear anisotropic diffusion equations. Special attention is devoted to finding the weakest integrability assumption for the right-hand side which allows the arguments to be carried out.

H\{o}lder Regularity for Doubly Nonlinear Equations

Yevgeniia Yevgenieva Max Planck Institute for Dynamics of Complex Technical Systems Germany Co-Author(s):    Simone Ciani, Eurica Henriques, Mariia Savchenko, Igor Skrypnik

Abstract: We study the local H\{o}lder continuity of nonnegative solutions to the doubly nonlinear parabolic equation \begin{equation*} u_t-\textrm{div} \big(|Du^m|^{p-2} Du^m \big)=0 \end{equation*} in the mixed degenerate-singular cases, up to certain Barenblat numbers: \begin{equation*} 01,\quad \frac{2N}{N+1}