Special Session 71: Pure and Applied Analysis, Local and Nonlocal

Harmonic functions are Lipschitz continuous

Karthik Adimurthi TIFR-CAM India Co-Author(s):    Karthik Adimurthi

Abstract: In this talk, we discuss some old ideas from DeGiorgi-Nash-Moser theory and Piccinini-Spagnolo. Subsequently, we propose a possible strategy to extend the result of Piccinini-Spagnolo to higher dimensions and prove this idea for harmonic functions.

A global Calderon-Zygmund theory for nonlocal elliptic equations

Sun-Sig Byun Seoul National University Korea Co-Author(s):    Sun-Sig Byun

Abstract: In this talk a global Calderon-Zygmund theory is discussed, focusing particularly on a nonlocal elliptic equation with discontinuous coefficients on a bounded domain.

Pasting embeddings of pieces

Florin Catrina St. John`s University USA Co-Author(s):    S. Ostrovska, M. Ostrovskii

Abstract: One of the local-global themes in the theory of metric embeddings is: suppose that all bounded subsets of an unbounded metric space $A$ admit bilipschitz embeddings into a Banach space $X$ with uniformly bounded distortions. Does the whole metric space $A$ admit a bilipschitz embedding into $X$? In some cases, we answer this question positively by using smooth transitions between the parts` embeddings; the construction is based on logarithmic spirals.

Asymptotic behaviour of three fractional spaces

Ahmed Dughayshim University of Pittsburgh USA Co-Author(s):

Abstract: We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a stability theory a la Bourgain-Brezis-Mironescu as $s \to 1$, answering a question raised by Brazke--Schikorra--Yung. Part of the results are new even for $p=q$.

Fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity

Siran Li Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Xiangxiang Su

Abstract: A fundamental result in global analysis and nonlinear elasticity asserts that given a solution $\mathfrak{S}$ to the Gauss--Codazzi--Ricci equations over a simply-connected closed manifold $(\mathcal{M}^n,g)$, one may find an isometric immersion $\iota$ of $(\mathcal{M}^n,g)$ into the Euclidean space $\mathbb{R}^{n+k}$ whose extrinsic geometry coincides with $\mathfrak{S}$. Here the dimension $n$ and the codimension $k$ are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on $\mathfrak{S}$ and $\iota$. The best result up to date is $\mathfrak{S} \in L^p$ and $\iota \in W^{2,p}$ for $p>n \geq 3$ or $p=n=2$. In this paper, we extend the above result to $\iota \in \mathcal{X}$ whose topology is strictly weaker than $W^{2,n}$ for $n \geq 3$. Indeed, $\mathcal{X}$ is the weak Morrey space $L^{p, n-p}_{2,w}$ with arbitrary $p \in ]2,n]$. This appears to be first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss--Codazzi--Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges --- in particular, Rivi\`{e}re--Struwe`s work on harmonic maps in arbitrary dimensions and codimensions --- and compensated compactness.

Variational Analysis of a Parametrized Family of Transmission Problems Coupling Nonlocal and Fractional Models

Tadele Mengesha The University of Tennessee, Knoxville USA Co-Author(s):

Abstract: I will present a recent work on the analysis of a parameterized family of energies associated with transmission problems that effectively couple two distinct models across an interface. Specifically, we examine the coupling between a model based on the regional fractional Laplacian and another model employing a nonlocal operator with a position-dependent interaction kernel. Both operators are inherently nonlocal and act on functions defined within their respective domains. The coupling occurs via a transmission condition across a hypersurface interface. The heterogeneous interaction kernel of the nonlocal operator leads to an energy space endowed with a well-defined trace operator. This, combined with well-established trace results of fractional Sobolev spaces, facilitates the imposition of a transmission condition across an interface. The family of problems will be parametrized by two key parameters that measure nonlocality and differentiability. For each pair of parameters, we demonstrate existence of a solution to the resulting variational problems. Furthermore, we investigate the limiting behavior of these solutions as a function of the parameters.

Partial regularity in nonlocal systems

Simon Nowak Bielefeld University Germany Co-Author(s):    Cristiana De Filippis, Giuseppe Mingione

Abstract: The theory of partial regularity for elliptic systems replaces the classical De Giorgi-Nash-Moser theory for scalar equations, asserting that solutions are regular outside of an in general non-empty negligible closed subset called the singular set. The local theory was initiated by Giusti \& Miranda and Morrey, in turn relying on De Giorgi`s seminal ideas in the context of minimal surfaces. I will present several extensions of the classical local partial regularity theory to nonlinear integro-differential systems along with some general tools for proving $\varepsilon$-regularity theorems in nonlocal settings. This is joint work with Cristiana De Filippis and Giuseppe Mingione (Parma).

A weighted Schauder estimate for an irregular oblique derivative problem

Michiaki Onodera Tokyo Institute of Technology Japan Co-Author(s):    Toru Kan, Rolando Magnanini

Abstract: I will talk about an elliptic regularity estimate for an oblique derivative boundary value problem, in which the directional derivative of solution along a vector field on the boundary is prescribed. It is known that the classical Schauder estimate is no longer valid if the vector field is tangential at a submanifold of the boundary. Our new Schauder-type estimate, in contrast to previously known subelliptic estimates, does not lose the derivatives, but it takes into account the regularity deficit as a weight in the estimate. In fact, this estimate is shown to be appropriate for an application to Backus` problem in geodesy. This talk is based on joint work with Toru Kan (Osaka Metropolitan University) and Rolando Magnanini (University of Florence).

Nonlocal boundary-value problems with local boundary conditions

James Scott Columbia University USA Co-Author(s):    Qiang Du

Abstract: We state and analyze nonlocal problems with classically-defined, local boundary conditions. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We establish a Green`s identity for the nonlocal operator that recovers the classical boundary integral, which permits the use of variational techniques. Using this, we show the existence of weak solutions, as well as their variational convergence to classical counterparts as the horizon uniformly converges to zero. In certain circumstances, global regularity of solutions can be established, resulting in improved modes and rates of variational convergence. We also show that Galerkin discretization schemes for the nonlocal problems converge unconditionally with respect to the nonlocal parameter, i.e. that the schemes are asymptotically compatible.

Nonlocal Sublinear Elliptic Problems with Measure Coefficients and Data

Adisak Seesanea Sirindhorn International Institute of Technology, Thammasat University Thailand Co-Author(s):

Abstract: We study elliptic equations of the form \( (-\Delta)^{\frac{\alpha}{2}} u = f(x,u)\) in \(\mathbb{R}^{n}\), where \((-\Delta)^{\frac{\alpha}{2}}\) denotes the fractional Laplacian in \(\mathbb{R}^{n}\) for \( 0 < \alpha < n \) and \(n \geq 2\). The nonlinearity \(f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega\) includes sublinear growth terms, with \( 0 < q_i < 1\), the coefficients \(\sigma_{i}\) and the data \(\omega\) are Radon measures on \(\mathbb{R}^n\). We will present results on the existence, uniqueness, and pointwise estimates for some classes of solutions to these problems. This talk is based on joint work with Kentaro Hirata, Aye Chan May, Toe Toe Shwe, and Igor E. Verbitsky.

Improved moduli of continuity for degenerate phase transitions

José Miguel Urbano King Abdullah University of Science and Technology (KAUST) Saudi Arabia Co-Author(s):    Ugo Gianazza and Naian Liao

Abstract: We substantially 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.