Special Session 107: Recent advances in regularity theory for local and nonlocal elliptic and parabolic equations

$\mu$-ellipticity and nonautonomous integrals

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

Abstract: $\mu$-ellipticity describes certain degenerate forms of ellipticity typical of convex integrals at linear or nearly linear growth, such as the area integral or the iterated logarithmic model. The regularity of solutions to autonomous or totally differentiable problems is classical after Bombieri, De Giorgi and Miranda, Ladyzhenskaya and Ural`tseva and Frehse and Seregin. The anisotropic case is a later achievement of Bildhauer, Fuchs and Mingione, Beck and Schmidt and Gmeineder and Kristensen. However, all the approaches developed so far break down in presence of nondifferentiable ingredients. In particular, Schauder theory for certain significant anisotropic, nonautonomous functionals with Holder continuous coefficients was only recently obtained by C. De Filippis and Mingione. We will see the validity of Schauder theory for anisotropic problems whose growth is arbitrarily close to linear within the maximal nonuniformity range, and discuss sharp results and insights on more general nonautonomous area type integrals. We also present an intrinsic approach to Schauder theory for general nonautonomous functionals at nearly linear growth, covering the most relevant model examples in the literature. From recent joint work with Cristiana De Filippis (Parma), Mirco Piccinini (Polimi) and Peter Hasto (Helsinki).

Existence results for nonlinear diffusion equations with drifts in divergence form

Sukjung Hwang Chungbuk National University Korea Co-Author(s):

Abstract: In this presentation, we discuss recent existence results for nonlinear diffusion equations with drift terms in divergence form. These results are broadly applicable to a variety of reaction diffusion models, including Keller Segel models. We emphasize the identification of functional spaces for the drift term, guided by the diffusion nonlinearity and the regularity of the initial data.

Degenerate Nonlinear Partial Differential Equations in Curvature Flows

Ki-Ahm Lee Seoul National University Korea Co-Author(s):

Abstract: This talk studies degenerate nonlinear partial differential equations arising in curvature flows with noncompact graphical initial hypersurfaces. The evolving hypersurface remains graphical and develops infinite height along the boundary of its support, leading naturally to a free boundary formulation. We establish uniform a priori estimates up to the infinite-height boundary despite the degeneracy of the equation, and analyze the evolution of the support and its free boundary. Geometric quantities preserved under the flow are also examined. Finally, we discuss cigar-type traveling wave solutions with flat regions to address Hamilton's problem on the existence of such flows.

Liouville-type theorems for nonlocal Lane--Emden inequalities via test function methods

Taehun Lee Konkuk University Korea Co-Author(s):    Takwon Kim

Abstract: We present Liouville-type theorems for nonnegative weak supersolutions of nonlocal Lane--Emden inequalities driven by integro-differential operators of order $2s$ with $s\in(0,1)$. In the linear setting, we consider $L_K u = u^q$ in $\mathbb{R}^n$, where $K$ is an even, uniformly elliptic kernel, and prove that $u\equiv 0$ for $1

Sobolev flow and doubly nonlinear parabolic type equations

Masashi Misawa Kumamoto university Japan Co-Author(s):

Abstract: In this talk we shall study the gradient flow associated with the Sobolev inequality, called the p-Sobolev flow, which is described as a doubly nonlinear degenerate and singular parabolic equation. In the case p=2 the p-Sobolev flow is related to the so-called Yamabe flow in differential geometry. We consider Cauchy-Dirichlet problem for the p-Sobolev flow, and we show a boundedness, a positivity, a regularity of the solution and the asymptotic behavior at infinite-time of the p-Sobolev flow. For the proof we study a local boundedness, the so-called positivity-expansion being valid for doubly nonlinear parabolic equations. Our global existence of the p-Sobolev flow is based on the scaling transformation intrinsic to the doubly nonlinear parabolic equation and this approach also eventually leads to an application to the finite-time extinction-behavior for the so-called fast diffusive doubly nonlinear parabolic equation. This is partially based on a collaborative work with Tuomo Kuusi in University of Helsinki, Finland and Kenta Nakamura in Kumamoto University. References: T. Kuusi, M. Misawa, K. Nakamura: J. Geom. Anal. 30 no.2 (2020); J. Differ. Eq. 279 (2021); M. Misawa, K. Nakamura: Adv. Calc. Var. (2021); J. Geom. Anal. 33 no.1: 33 (2023); M. Misawa, K. Nakamura, Md Abu Hanif Sarkar: Nonlinear Differ. Eqn. Appl. 30 no.3: 43 (2023); M. Misawa: Calc. Var. 62 (2023), no. 9: 265; M. Misawa, Y. Yamaura: Math. Annalen (2025); T. Kuusi, M. Misawa, K. Nakamura: submitted

Calder\`{o}n-Zygmund estimates for $(s,p)$-Poisson type equations

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

Abstract: We discuss gradient estimates of Calder\`{o}n-Zygmund type for local weak solutions to fractional $p$-Poisson type equations of order $s$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$, these estimates lead to optimal integrability of the weak gradient in terms of the integrability of the right-hand side.

Conormal Derivative Problems for Quasilinear Elliptic Equations with Morrey Data

Dian K Palagachev Politecnico di Bari Italy Co-Author(s):

Abstract: We will discuss regularity issues regarding the weak solutions to non-homogeneous conormal derivative problem for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carath\`{e}odory functions and satisfy controlled growth structure conditions with respect to the solution and its gradient, while their $x$-behaviour is controlled in terms of suitable Morrey spaces. Global boundedness up to the boundary will be shown for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural`tseva to the framework of the Morrey scales.

Regularity theory for parabolic double phase problems

Abhrojyoti Sen Goethe University Frankfurt Germany Co-Author(s):

Abstract: In this talk, we discuss recent regularity results for parabolic double-phase equations and systems, focusing on Lipschitz regularity in the presence of gradient nonlinearities. Our approach is based on the celebrated Ishii-Lions method for viscosity solutions. We show that when the modulating coefficient is spatially Lipschitz and its zero set satisfies a mild and natural control condition, then bounded weak solutions to parabolic double-phase equations with gradient nonlinearity are locally Lipschitz continuous in space and 1/2 H\{o}lder continuous in time. Furthermore, we also discuss the higher integrability result for parabolic double phase systems having two modulating coefficients.

Generalized Structural Conditions and Gradient Regularity for Double Phase Problems

Pilsoo Shin Kyonggi University Korea Co-Author(s):    Yeonghun Youn

Abstract: In this talk, I introduce a generalized structural condition for the double phase problem. The main operator is expressed as the diagonal component of a suitable vector field $G(x, y, z)$ involving two spatial variables, where $x$ is the variable modulating the growth and $y$ is the variable that allows some additional irregularity on the main operator. Under this generalized structure, we show that by assuming an appropriate regularity assumption on the $y$ variable, one can obtain certain regularity of the gradient of solutions.

Approximation of the Solutions to Quasilinear Parabolic Problems with Perturbed Coefficients

Lyoubomira Softova Salerno University Italy Co-Author(s):    Rosamaria Rescigno

Abstract: We consider the Cauchy-Dirichlet problem for second order quasilinear operators of parabolic type in non-divergence form. The data are Carath{\`e}odory functions, and the principal part is of $VMO_x$-type with respect to the variables $ (x,t).$ Assuming the existence of a strong solution $u_0,$ we apply the Implicit Function Theorem in a neighbourhood of this solution to show that small bounded perturbations of the data lead to small perturbations of the solution $u_0$ itself. Furthermore, we employ the Newton iteration procedure to construct an approximating sequence that converges to $u_0$ in the corresponding Sobolev space.