Schedule for: 21w5100 - Nonlinear Potential Theoretic Methods in Partial Differential Equations (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions $10$ and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension $9$. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

We will survey the regularity theory of minimizers of the Mumford-Shah functional, focusing in particular on that of the corresponding singular sets. Starting with nowadays classical results, we will finally discuss more recent developments

We obtain the boundedness of the Hardy-Littlewood maximal function on $L^q$ type spaces defined via Choquet integrals associate to Sobolev capacities. The bounds are obtained in full range of exponents including a weak type end-point bound. We also obtain a capacitary inequality of Maz'ya type which resolves a problem proposed by D. Adams. This talk is based on joint work with Keng Hao Ooi.

We discuss the role of entropy functionals played in the study of Gauss curvature type flows: 1. the monotonicity of the associated entropies, 2. diameter, non-collapsing entropy points estimates, 3. convergence. Similar entropy functionals also exists for anisotropy type Gauss curvature flows.

The Bernstein problem asks whether entire minimal graphs in dimension N+1 are necessarily hyperplanes. This problem was solved in combined works of Bernstein, Fleming, De Giorgi, Almgren, and Simons ("yes" if N < 8), and Bombieri-De Giorgi-Giusti ("no" otherwise). We will discuss the analogue of this problem for graphical minimizers of anisotropic energies. In particular, we will discuss new examples of nonlinear entire graphical minimizers in the case N = 6, and recent joint work with Y. Yang towards constructing such examples in the lowest-possible-dimensional case N = 4.

We consider measure data elliptic problems involving a second order operator exhibiting Orlicz growth and having measurable coefficients. As known in the $p$-Laplace case, pointwise estimates for solutions expressed with the use of nonlinear potential are powerful tools in the study of the local behaviour of the solutions. Not only we provide such estimates expressed in terms of a potential of generalized Wolff type, but also we investigate their regularity consequences. For scalar equations we do not need to impose any structural conditions on the the operator and we study generalized $A$-harmonic functions being distributional solutions to problems with nonnegative measure. Lower and upper estimates we provide are sharp in the sense that the potential cannot be substituted with a better one. As a consequence we get a bunch of sharp criteria for continuity or H\"older continuity of the solutions. For systems we impose typical assumptions of the Uhlenbeck-type structure of the operator and separated variables, whereas the measure can be signed as another notion of very weak solutions is employed. In this case the upper bound is shown with the same potential as in the scalar case and presented together with its precise consequences for the local behaviour of solutions. The talk is based on joint works:(scalar) with F.~Giannetti and A.~Zatorska-Goldstein [arXiv:2006.02172] and (vectorial) with Y.~Youn and A.~Zatorska-Goldstein, [arXiv:2102.09313], [arXiv:2106.11639].

We discuss the construction of a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains. This is a joint work with Olli Saari.

We will consider nonlinear, uniformly elliptic equations with variational structure and random, highly oscillating coefficients and discuss the corresponding stochastic homogenization theory. After recalling basic ideas on how to get quantitative rates of homogenization for nonlinear uniformly convex problems, we will discuss our recent work, jointly with S. Armstrong and S. Ferguson, showing that homogenization and linearization commute. This is in the sense that the linearized equation homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). This procedure can be iterated to show higher regularity of the homogenized Lagrangian as well as large-scale regularity for minimizers.

We consider an elliptic system that is motivated by a congested traffic dynamics problem. It has the form $$\mathrm{div}\bigg((|Du|-1)_+^{p-1}\frac{Du}{|Du|}\bigg)=f,$$ and falls into the context of very degenerate problems. Continuity properties of the gradient have been investigated in the scalar case by Santambrogio & Vespri and Colombo & Figalli. In this talk we establish the optimal regularity of weak solutions in the vectorial case for any $p>1$. This is joint work with F. Duzaar, R. Giova and A. Passarelli di Napoli.

In this talk we consider a large class of non-uniformly elliptic variational problems and discuss optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the data. The analysis covers the main model cases of variational integrals of anisotropic growth, but also of fast growth of exponential type investigated in recent years. The regularity criteria are established by potential theoretic arguments, involve natural limiting function spaces on the data, and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. The results presented in this talk are part of a joined project with Giuseppe Mingione

Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally Hoelder continuous, provided coefficients are locally Hoelder continuous. In this talk I will present new regularity results for minima of nonuniformly elliptic functionals with emphasis on delicate borderline regulairty criteria. My talk is based on papers: -C. De Filippis, Quasiconvexity and partial regularity via nonlinear potentials. Preprint (2021); -C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals. Arch. Ration. Mech. Anal., to appear; C. De Filippis, G. Mingione, Nonuniformly elliptic Schauder estimates. Preprint (2021).

For a positive integer $N$, the $N$-membranes problem describes the equilibrium position of $N$ ordered elastic membranes subject to forcing and boundary conditions. If the heights of the membranes are described by real functions $u_1, u_2,...,u_N$, then the problem can be understood as a system of $N-1$ coupled obstacle problems with interacting free boundaries which can cross each other. When $N=2$ there is only one free boundary and the problem is equivalent to the classical obstacle problem. I will discuss a work in collaboration with Hui Yu about the regularity of the free boundaries in the two dimensional case.

Motivated by the question from mathematical physics about the size of magnetic fields that support zero modes for the three dimensional Dirac equation, we study a certain conformally invariant spinor equation. We state some conjectures and present results in their support. Those concern, in particular, two novel Sobolev inequalities for spinors and vector fields. The talk is based on joint work with Michael Loss.

It is a convenient and well-known fact that for exponents p>1, any Lp-weakly converging sequence of PDE constrained maps admits a decomposition into sequences of PDE constrained maps where one converges in measure (no oscillation) and the other is p-equi-integrable (no concentration). For p=1 the relevant corresponding result concerns weakly* convergent sequences of PDE constrained measures and is false: the oscillation and concentration cannot be separated while simultaneously satisfying the PDE constraint. In this talk we explain how the concentration regardless of the failure of a decomposition result retains its PDE character. The presented results are parts of joint works with Andre Guerra and Bogdan Raita.

The talk will be based on some papers in collaboration with Kazuhiro Ishige (The University of Tokyo) and Asuka Takatsu (Tokyo Metropolitan University) where we investigate the preservation of concavity properties by heat flow. Surprisingly, we have recently proved that there exist concavities stronger than log-concavity that are preserved by the Dirichlet heat flow, however, when we consider a suitable class of concavities, log-concavity remains the strongest possible. Moreover, in our latest paper, we prove that, when starting with an initial datum which shares any concavity weaker than log-concavity, then the solution may lose immediately any reminiscence of concavity. In this way we almost complete the study of preservation of concavity by the Dirichlet heat flow, started by Brascamp and Lieb in 1976.

We show how p-harmonic functions and Sobolev spaces on metric spaces, based on upper gradients, naturally lead to fine potential theory, even in the setting of Euclidean spaces.

I will discuss a new estimate, obtained in joint work with M. M. Fall, P.A. Feulefack and R.Y. Temgoua, on the Morse index of radially symmetric sign changing solutions to semilinear fractional Dirichlet problems in the unit ball. In particular, the result applies to the Dirichlet eigenvalue problem for the fractional Laplacian and implies that eigenfunctions corresponding to the second Dirichlet eigenvalue are antisymmetric. This resolves a conjecture of Banuelos and Kulczycki.

I will discuss recent work with T. Kuusi and C. Smart on quantitative unique continuation for solutions of periodic elliptic equations on large scales.

We prove global Besov estimates for the p-Laplacian with right-hand side in divergence form under optimal assumptions on the regularity of the boundary of the domain $\Omega$. In particular, we show that $B^s_{\varrho,q}(\Omega)$-regularity transfers from the forcing $F$ to the non-linear flux $|\nabla u|^{p-2}\nabla u$ provided the boundary belongs to the class $B^{s+1-1/q}_{\varrho,q}$ and has a small Lipschitz constant. In the linear case $p=2$ this recovers a sharp result from Maz'ya-Shaposhnikova. This is a joint work with A. Balci and L. Diening.

Given $\Omega$ a subdomain of the Poincaré ball $\mathbb{B}_n$, we investigate properties of variational solutions $u: \Omega\to\mathbb{R}$ to the borderline Dirichlet problem, $$\begin{eqnarray} \left\{ \begin{array}{lll} -\Delta_{\mathbb{B}_n}u-\gamma{V_2}u -\lambda u&=V_{2^\star(s)}|u|^{2^\star(s)-2}u &\hbox{ in }\Omega\\ \hfill u &=0 & \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray}$$ where $\gamma \leq \frac{(n-2)^2}{4}$, $0< s <2$, $2^\star(s):=\frac{2(n-s)}{n-2}$ being the corresponding critical Sobolev exponent, while $V_{2},V_{2^\star(s)}$ are weights that make the problem invariant under hyperbolic scaling. In particular, $V_2$ behaves like the classical Hardy-Weight $r^{-2}$ around $0$. Via a sharp blow-up analysis, we give conditions for stability and compactness of solutions to the above problem. These conditions rely on a « mass » in low-dimensions. This allows to get existence and multiplicity of solutions. We also get classical Pohozaev-type non-existence results: our blow-up analysis yields the stability of these obstructions in low dimension. This is joint work with Nassif Ghoussoub (UBC) and Saikat Mazumdar (IIT Bombay).

We consider a partially damped nonlinear beam-wave system of evolution PDE's modeling the dynamics of a degenerate plate. The plate can move both vertically and torsionally and, consequently, the solution has two components. We show that the component from the damped beam equation always vanishes asymptotically while the component from the (undamped) wave equation does not. In case of small energies we show that the first component vanishes at exponential rate. Our results highlight that partial damping is not enough to steer the whole solution to rest and that the partially (controlled) damped system can be less stable than the undamped system. Hence, the model and the behavior of the solution enter in the framework of the so-called indirect damping and destabilization paradox. These phenomena are valorized by a physical interpretation leading to possible new explanations of the Tacoma Narrows Bridge collapse and to possible damages due to the damping control parameter. This is joint work with A. Soufyane (Sharjah, UAE), based on a previous model developed with M. Garrione (Milano, Italy).

In this talk I will present a result on Sobolev regularity of weak solutions to linear nonlocal equations. The theory we develop is concerned with obtaining higher integrability and differentiability of solutions of linear nonlocal equations. In addition to the standard conditions on the coefficient symmetricity and ellipticity, if we assume uniformly Holder continuity of the coefficient, then weak solutions from the energy space that correspond to highly integrable right hand side will have an improved Sobolev regularity along the differentiability scale in addition to the expected integrability gain. This result is consistent with self-improving properties of nonlocal equations that has been observed by other earlier works. To prove our result, we use a perturbation argument where optimal regularity of solutions of a simpler equation is systematically used to derive an improved regularity for the solution of the nonlocal equation. This is a joint work with Armin Schikorra and Sasikarn Yeepo.

We obtain new local Calderon-Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as non-linear equations. We introduce a novel $\log-BMO$ condition on the weight. In particular, we assume smallness of the logarithm of the matrix-valued weight in $BMO$. This allows to include degenerate, discontinuous weights. The assumption on the smallness parameter is sharp and linear in terms of the integrability exponent of the gradient. This is a novelty even in the linear setting with non-degenerate weights compared to previously known results, where the dependency was exponential. We also consider regularity up to the boundary. The exponent of integrability depends again linearly on the smallness condition on the boundary.

We discuss conditions for Lipschitz and $C^1$ regularity for a uniformly elliptic equation in divergence form with coefficients that were introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous. (This is joint work with V.G.Maz’ya.)

For every summable function $f$, we prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems in a bounded open subset $\Omega$ of $\mathbb R^N$. The principal part is a divergence-form nonlinear anisotropic operator $\mathcal A$, the prototype of which is $$\mathcal A u=-\sum_{j=1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u)$$ with $p_j>1$ for all $1\leq j\leq N$ and $\sum_{j=1}^N (1/p_j)>1$. As a novelty, our lower order terms involve a new class of operators $\mathfrak B$ such that $\mathcal{A}-\mathfrak{B}$ is bounded, coercive and pseudo-monotone from $W_0^{1,\overrightarrow{p}}(\Omega)$ into its dual, as well as a gradient-dependent nonlinearity with an ``anisotropic natural growth" in the gradient and a good sign condition. This is joint work with Barbara Brandolini (Universita degli Studi di Palermo, Italy).

In this talk we consider perturbations of critical stationary Schrodinger equations, such as Yamabe-type equations on manifolds or Brézis-Nirenberg-type equations on bounded open sets. We are interested in the blow-up behavior of such equations; in particular in how blowing-up solutions may develop « multi-bubble blow-up », that is how several interacting concentrating peaks may appear. In dimensions larger than 7, on a locally conformally flat manifold, we construct positive blowing-up solutions of such equations that behave like towers of bubbles concentrating at a critical point of the mass function. The result does not assume any symmetry on the underlying manifold. The construction is performed by combining finite-dimensional reduction methods with a linear bubble-tree analysis. Our approach works both in the positive and sign-changing case: as a byproduct of our analysis we prove the existence, on a generic bounded open set of $\mathbb{R}^n$, of blowing-up solutions of the Brézis-Nirenberg equation that behave like towers of bubbles of alternating signs.

Ginzburg–Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau–de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group. We obtain similar results for $p$–harmonic maps with $p<2$ going to $2$. The results unify the existing theory and cover new situations and problems. This is a joint work with Antonin Monteil (Paris-Est Créteil, France), Rémy Rodiac (Paris–Saclay, France) and Benoît Van Vaerenbergh (UCLouvain).

Optimal insulation consists in finding the ``best" displacement of a prescribed volume of insulating material around a given conductor. According to circumstances, the ``best" configuration can be the one which minimizes the heat dispersion, maximizes the heat content, minimizes the heat rate loss etc. We provide a flavor of the state of the art, and then we focus on the case of prescribed heat source (inside the conductor), with convective heat transfer across the solid and the environment. This corresponds to consider the stationary heat equation inside both conductor & insulator together with Robin boundary conditions at the external boundary. We aim at maximizing the heat content (the $L^1$ norm of the solution) among all the possible distributions of insulating material with fixed mass, and we prove an optimal upper bound in terms of geometric quantities alone. Eventually we prove a conjecture according to which the ball surrounded by a uniform distribution of insulating material maximizes the heat content.

Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\mathbb R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.This is a joint work with D. Jerison and H. Shahgholian.

The Onsager conjecture, recently solved by Phil Isett, states that, below a certain threshold regularity, Hoelder continuous solutions of the Euler equations might dissipate the kinetic energy. The original work of Onsager was motivated by the phenomenon of anomalous dissipation and a rigorous mathematical justification of the latter should show that the energy dissipation in the Navier-Stokes equations is, in a suitable statistical sense, independent of the viscosity. In particular it makes much more sense to look for solutions of the Euler equations which, besides dissipating the total kinetic energy, satisfy as well a suitable form of local energy inequality. Such solutions were first shown to exist by Laszlo Szekelyhidi Jr. and myself. In this talk I will review the methods used so far to approach their existence and the most recent results by Isett and by Hyunju Kwon and myself.