TWO DAYS ON REGULARITY RESULTS

FOR VARIATIONAL PROBLEMS AND PDES

Modena, 2-3 December 2021

Giornate di lavoro nell’ambito del Progetto GNAMPA 2020 Regolarità ottimale, di ordine intero e frazionario, per problemi con ostacolo sotto ipotesi di crescita generali

Program


December, 2nd, 2021

Aula M2.2, Edificio Matematica
chair Sergio Polidoro
15:00 — 15:20Chiara Gavioli (online: meet.google.com/hrv-uuvf-fkc)

Higher differentiability for a class of obstacle problems with non-standard growth conditions

Abstract – We establish the higher differentiability of integer order of the solutions to a class of obstacle problems of the form
$$\min \left\lbrace \int_\Omega F(x,Dw)\, {\rm d}x : w \in \mathcal{K}_{\psi}(\Omega) \right\rbrace,$$
assuming that the gradient of the obstacle possesses an extra integer differentiability property. The energy density $F$ is assumed to satisfy suitable $p,q$-growth conditions with $p$ and $q$ linked by the relation
\begin{equation}
\frac q p < 1 + \frac 1 n – \frac 1 r\,, \qquad \qquad \qquad \qquad (1)
\end{equation}
being $r > n$. Here $\psi \in W^{1,p}(\Omega)$ is a fixed function called obstacle for which we assume $\nabla\psi \in W^{1,2q-p}_{\rm loc}(\Omega)$, and $\mathcal{K}_{\psi}(\Omega) = \{w \in W^{1,p}(\Omega): w \ge \psi \,\, \textnormal{a.e. in }\Omega\}$ is the class of admissible functions. We require for the partial map $x \mapsto D_\xi F(x,\xi)$ a higher differentiability of Sobolev order in the space $W^{1,r}$, with $r > n$ from condition (1).

This result can be used to obtain Lipschitz continuity for a class of obstacle problems under non-standard growth conditions.
15:20 — 15:40Andrea Scapellato

Recent advances on some integral operators in Morrey spaces with mixed norm

Abstract – Morrey spaces were introduced by C. Morrey to examine the local behavior of solutions to second-order elliptic partial differential equations. In the last years, a lot of Morrey-type spaces have been studied.
In this talk, we present some recent results related to Morrey spaces with mixed norm.
Let $\Omega \subset \mathbb{R}^n$ be a bounded set, $B_\rho(x)$ be the ball centered at $x\in\mathbb{R}^n$ and with radius $\rho$, $1 < p , q < + \infty, 0 < \lambda < n, 0 < \mu < 1$ and fix $T>0$. Then, following [1,2,3] we define the Morrey space with mixed norm $L^{q,\mu}(0,T, L^{p,\lambda}(\Omega)) $ as the class of all functions $f:\Omega \times (0,T)\to \mathbb{R}$ such that the norm
$$
\|f\|_{L^{q,\mu}(0,T,L^{p,\lambda}(\Omega))} :=
\left( \sup_{t_0 \in (0,T) \atop \rho>0}
\frac{1}{\rho^\mu}
\int\limits_{(0,T) \cap (t_0-\rho, t_0+\rho)}\!\!\!
\left(
\sup_{x \in \Omega \atop \rho > 0}
\frac{1}{\rho^\lambda}
\int\limits_{\Omega\cap B_\rho (x)} |f(y,t)|^p\,\mathrm{d}y
\right)^{\!\!\frac{q}{p}}
\mathrm{d}t\,
\right)^{\!\!\frac{1}{q}}
$$
is finite.
We discuss some boundedness results for fractional integral operators and their commutators with functions with bounded mean oscillation and we investigate some regularity results for parabolic equations assuming that the coefficients of the parabolic operator belong to the Sarason class of functions with vanishing mean oscillation. Furthermore, we show some new results related to fractional integrals associated to operators with Gaussian kernel bounds, announced in [2]. Some open problems concerning the regularity theory in Morrey spaces with mixed norm are discussed.

[1] M.A. Ragusa and A Scapellato, Mixed Morrey spaces and their applications to partial differential equations, Nonlinear Anal.-Theory Methods Appl. 151 51-65 (2017).
[2] F. Anceschi, C.S. Goodrich, A. Scapellato, Operators with Gaussian kernel bounds on Mixed Morrey spaces, Filomat, 33(16), 5219-5230 (2019).
[3] A. Scapellato, A Modified Spanne-Peetre Inequality on Mixed Morrey Spaces, Bull. Malays. Math. Sci. Soc., 43(6), 4197-4206 (2020).
15:40 — 16:00Annalaura Rebucci

On the weak regularity of solutions to degenerate Kolmogorov equations

Abstract – We here discuss the weak regularity of solutions to degenerate Kolmogorov equations as we studied it in [2]. More precisely, we prove a Harnack inequality and the Hölder continuity for weak solutions to equation $\mathscr{L} u=f$, under the assumptions of measurable coefficients, integrable lower order terms and nonzero source term.
Our proof relies on the combination of three fundamental ingredients – boundedness of weak sub-solutions, weak Poincaré inequality and Log-transformation – in the same spirit of the recent paper [3] for the Fokker-Planck equation. In particular, we prove the boundedness of weak sub-solutions to equation $\mathscr{L} u= f$ using the Moser’s iterative method. Thus, we manage to lower the integrability assumption on the lower order coefficients and to handle a non-vanishing source term $f$, extending to this more general case the Moser’s iterative scheme proposed in [4] and subsequently in [1].
Finally, we introduce a function space $\mathcal{W}$, which is the most natural framework for the study of the weak regularity theory for operator $\mathscr{L}$ and allows us to prove a weak Poincaré inequality.

[1] F. Anceschi, S. Polidoro, and M. A. Ragusa, Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients, Nonlinear Analysis 1–19, (2019).
[2] F. Anceschi and A. Rebucci, A note on the weak regularity theory for degenerate Kolmogorov equations, preprint arXiv:2107.04441, (2021)
[3] J. Guerand and C. Imbert, Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations, preprint arXiv: 2102.04105, (2021)
[4] A. Pascucci and S. Polidoro, The Moser’s iterative method for a class of ultraparabolic equations, Commun. Contemp. Math. 6, 395–417, (2004).
16:00 — 16:20Mirco Piccinini

Nonlinear Fractional Equations in the Hesenberg group $\mathbb{H}^n$

Abstract – In the non-Euclidean setting of the Heisenberg group $\mathbb{H}^n$ we present some qualitative properties of the solution $u$ to the Dirichlet problem
\begin{equation}
\begin{cases}
\mathscr{L} u = f \quad \text{in} \quad \Omega \subset \mathbb{H}^n, \\
u = g \qquad \text{in} \quad \mathbb{H}^n \setminus \Omega,
\end{cases}
\qquad \qquad \qquad \qquad \qquad \qquad (1)
\end{equation}
where $f \equiv f( \cdot, u) \in L^\infty_{loc} (\mathbb{H}^n)$, $g \in W^{s,p}(\mathbb{H}^n)$ and $\mathscr{L}$ is an integro-differential operator of differentiability order $s \in (0, 1)$ and summability growth $p \in (1, \infty)$ whose prototype is the standard classical fractional subLaplacian in the Heisenberg group $\mathbb{H}^n (- \Delta_{\mathbb{H}^n} )^s$ defined as
$$
(- \Delta_{\mathbb{H}^n} )^s u(\xi) := c(n,s) \lim_{\delta \to 0} \int_{\mathbb{H}^n \setminus B_\delta(\xi)}
\frac{u(\xi) – u(\eta)}{|\eta^{-1} \circ \xi|_{\mathbb{H}^n}^{Q+sp}} d \eta, \qquad \xi \in \mathbb{H}^n
$$
with $| \cdot |_{\mathbb{H}^n}$ being the standard Heisenberg gauge.
We will extend the celebrated De Giorgi-Nash-Moser theory to the nonlinear and fractional operator $\mathscr{L}$ . In particular, we show that weak solutions to Problem (1) enjoy some properties such as boundedness, Hölder continuity and Harnack inequality. We will also establish general estimates such as a fractional Caccioppoli-type inequality with tail and a fractional Logarithmic Lemma.

[1] M. Manfredini, G. Palatucci, M. Piccinini, S. Polidoro: Hölder continuity and boundedness estimates for nonlinear fractional equations in the Heisenberg group. Submitted (2021).
[2] G. Palatucci, M. Piccinini: Harnack inequalities for nonlinear fractional equations in the Heisenberg group. Submitted (2021).

Coffee break

17:00 — 17:20Antonio Giuseppe Grimaldi

Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

Abstract – We establish the higher fractional differentiability properties of the gradient of solutions to a class of obstacle problems with non-standard growth conditions of $p, q$-type of the form
$$
\min \left\{ \int_ \Omega f(x, Dv) d x \mid v \in \mathcal{K}_\psi (\Omega)\right\}
$$
where $\Omega$ is a bounded open subset of $\mathbb{R}^n, n \ge 2, \psi \in W^{1,p}(\Omega)$ is a fixed function called obstacle and $\mathcal{K}_\psi (\Omega) = \left\{ w \in W^{1,p}(\Omega) \mid w \ge \psi \ a.e. \ \text{in} \ \Omega \right\}$ is the class of admissible functions. Assuming that the gap $q/p$ satisfies a suitable smallness condition and that both the gradient of the obstacle and the partial map $x \mapsto D_\xi F(x,\xi)$ belong to some suitable Besov space, we are able to prove that a fractional differentiability property transfers to the gradient of the solution. The strategy is to establish the a priori estimates for minimizers of functionals with $p$-growth conditions and then, by applying a suitable approximation argument, to show that the a priori estimates still hold for the minimizer of the initial problem.
17:20 — 17:40Erica Ipocoana

Regularity results in the scale of Besov spaces to a class of double-phase obstacle problems

Abstract – We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form
$$
\min \left\{ \int_ \Omega F(x, w, Dw) d x \mid w \in \mathcal{K}_\psi (\Omega)\right\} .
$$
In particular, the energy density $F : \Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ is defined by
$$
F(x, w, z) = b(x,w) H(x,z) = b(x,z) (|z|^p + a(x) |z|^q),
$$
where $\Omega$ is a bounded open subset of $\mathbb{R}^n, \psi \in W^{1,p}(\Omega)$ is a fixed function called obstacle and $\mathcal{K}_\psi (\Omega) = \left\{ w \in W^{1,p}(\Omega, \mathbb{R}) \mid w \ge \psi \ a.e. \ \text{in} \ \Omega \right\}$ is the class of admissible functions. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property. The main difficulty here is that the functional depends both on the $x$−variable and the $w$−variable. In order to overcome this issue, we adapt the strategy presented in [1]. Namely, we pass to a “freezed” obstacle problem for whom the results proved in [2] hold, so that we have that the solutions to the “freezed” obstacle problem inherit some Besov regularity of the obstacle.

[1] M. Eleuteri and A. Passarelli di Napoli, Regularity results for a class of non-differentiable obstacle problems Nonlinear Analysis 194, 111434 (2020)
[2] A.G. Grimaldi and E. Ipocoana, Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions, preprint arXiv:2109.01584 [math.AP] (2021).
17:40 — 18:00Andrea Torricelli

Regularity results for solutions of autonomous obstacle problems with general growth

Abstract – Considering the problem
$$
\min_{v \in \mathbb{K}_\psi(\Omega)} \int_{\Omega} F(Dv(x)) \, d x,
$$
our aim is to characterize the solutions by mean of a primal-dual formulation of the problem. We assume that the integrand satisfy some sort of convexity and super linearity at infinity. Our research is based on classical arguments of Convex Analysis like the convex approximation of function.
18:00 — 18:20Niccolò Foralli e Giovanni Giliberti

Higher differentiability of solutions for a class of obstacle problems with variable exponents

Abstract – We prove a higher differentiability result for the solutions to a class of obstacle problems in the form
$$
\min \left\{ \int_ \Omega f(x, Dw) d x \mid w \in \mathcal{K}_\psi (\Omega)\right\}
$$
where $\psi \in W^{1,p(x)}(\Omega)$ is a fixed function called obstacle and
$$
\mathcal{K}_\psi (\Omega) = \left\{ w \in W_0^{1,p(x)}(\Omega) + u_0 \mid w \ge \psi \ a.e. \ \text{in} \ \Omega \right\}
$$
is the class of the admissible functions, for a suitable boundary value $u_0$.
We deal with a convex integrand $F$ which satisfies the $p(x)$-growth conditions
$$
|\xi|^{p(x)} \le F(x,\xi) \le C( 1+ |\xi|^{p(x)}), \quad p(x) > 1.
$$

December, 3rd, 2021

Aula M1.6, Edificio Matematica
chair Claudia Capone
9:00 — 9:40Maria Alessandra Ragusa

Regularity for minimizers of some variational problems. Hints on existence results.

Abstract – In an open set $\Omega \subset \mathbb{R}^m~~(m\geq 2)$ let us define the maps $u: \Omega \to \mathbb{R}^n$. Also, let us consider the $p(x)$-energy functional as follows
$$
{\cal E}(u;\Omega):=\int_\Omega \Big(g^{\alpha\beta}(x)G_{ij}(u) D_\alpha u^i (x) D_\beta u^j (x) \Big)^{p(x)/2} dx,
$$
being $(g^{\alpha\beta}(x))$ and $(G_{ij}(u))$ symmetric positive definite matrices whose entries are continuous functions defined on $\Omega$ and $\mathbb{R}^n$ respectively, and $p(x)$ a continuous function on $\Omega$ with $p(x)\geq 2$.
Main focus is the study of regularity properties, interior and up to the boundary, of the minimizers $u$ of ${\cal E}$ and developments in this direction.
Some open problems concerning existence are discussed.
9:40 — 10:20Antonia Passarelli di Napoli

On the notion of solution to a class of problems with irregular obstacle

Abstract – We introduce the notion of very weak solution to a class of obstacle problem of the form
$$
\min_{u \in \mathcal{K}_\psi(\Omega)} \int_\Omega F(Du)
$$
where $\psi$ is a fixed function called obstacle and
$$\mathcal{K}_\psi(\Omega)= \left\{ v\in u_0+W^{1,r}_0(\Omega):\,\, v\ge \psi\,\, \text{a.e. in}\,\, \Omega \right\}
$$
is the class of the admissible competitors. We assume that $F(\xi)\approx |\xi|^p$ and that
$$\min\{1,p-1\}<r<p
$$
and discuss on the assumption on the obstacle needed to prove the existence of a very weak solution.
10:20 — 11:00Maria Manfredini

The role of mean value formulas in the study of the regularity for PDEs

Abstract – It is known that in the classical case of the Laplace operator the mean value formulas on the Euclidean spheres characterize the harmonic functions and are crucial in deriving their most salient properties: maximum principle (weak and strong), analyticity, Liouville’s theorem, Harnack’s inequality and so on.
For more general operators the level sets of the fundamental solution are a privileged class of ”balls”. This sets reflect the main properties of the operator, and in particular they give information on the directions of propagation of regularity, allowing to express in a natural and intrinsic way for example the Poincaré inequality and the potential theory results.
The first results extending the mean value formulas from the laplacian to the parabolic one are due to Pini and Watson and to sub-Riemannian setting are due to Citti, Garofalo and Lanconelli.
In this seminar we present some results related to the mean value formulas concerning fractional type operators and non linear operators. We consider the ”inverse” problem of mean value formulas and an obstacle problem. The link of mean value formulas with Poincaré inequality and with the Lebesgue approximation method for Perron’s solutions of the Dirichlet problem.

Coffee break


11:30 — 13:00

Round table


Lunch

15:00 — 15:20Andrea Gentile

Higher differentiability results for solutions to some non-homogeneouns elliptic problems

Abstract – We present a sharp higher differentiability result for local minimizers of functionals of the form
$$
\mathcal{F} = \int_{\Omega} [f (x, Dw(x)) − F(x) · w(x)] dx
$$
with non autonomous integrand $f(x, \xi)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1 < p < 2$. The main novelty here is that the results are obtained assuming that the partial map $x \mapsto D_{\xi}f(x, \xi)$ has weak derivatives in some Lebesgue space $L^
q$ and the datum $F$ is assumed to belong to a suitable Lebesgue space $L^r$.
We also prove that it is possible to weaken the assumption on the datum $F$ and on the map $x \mapsto D_{\xi}f(x, \xi)$, if the minimizers are assumed to be a priori bounded.
These results are contained in [1], a joint paper with A. Clop and A. Passarelli di Napoli.

[1] A. Clop, A. Gentile and A. Passarelli di Napoli. Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions, preprint (2021).
15:20 — 15:40Francesca Anceschi

On the Harnack inequality for the Kolmogorov equation with rough coefficients

Abstract – The Kolmogorov equation is a strongly degenerate second order pde, that was firstly introduced in 1934 as a fundamental ingredient for the description of the density of a system of $n$ particles of gas in the phase space. Later on, Hörmander considered it as a prototype for the family of hypoelliptic operators that can be written as a sum of squares.
Nowadays, the research community aims to study the weak regularity theory for this class of equations, by extending to this hypoelliptic framework the results that hold true in the elliptic and parabolic setting. However, due to the strong degeneracy of the equation, various precautions are required. In this talk, we recall the fundamental ideas behind the proof of a Harnack inequality for the Kolmogorov equations with rough coefficients and we characterize the set in which the estimate holds true. Lastly, we show some of its applications to the study of the fundamental solution associated to the Kolmogorov equation.
15:40 — 16:00Pasquale Ambrosio

Besov regularity for a class of singular or degenerate elliptic equations

Abstract – Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE
$$
-\mathrm{div}\left((\vert\nabla u\vert-1)_{+}^{q-1}\frac{\nabla u}{\vert\nabla u\vert}\right)=f,\,\,\,\,\,\mathrm{in}\,\,\Omega,
$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $1<q<\infty$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We assume that the datum $f$ belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of \textit{subquadratic growth}, i.e. $1<q<2$, which has so far been neglected. In the latter case, we also prove the higher fractional differentiability in the scale of Besov spaces of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case $q\geq2$.

Coffee break

16:30 — 16:50Samuele Riccò

Regularity for obstacle problems without structure conditions

Abstract –The aim of this seminar is to deal with the possible occurance of the Lavrentiev phenomenon on a variational obstacle problems with $p, q$−growth, when dealing with the Lipschitz continuity of solutions. In order to overcome this problem the availment of the notions of relaxed functional and Lavrentiev gap are needed. The main tool used here is a fundamental Lemma which reveals to be crucial because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions’ regularity that we intended to study. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients. This is a joint project in collaboration with Dr. Giacomo Bertazzoni (University of Modena and Reggio Emilia) and the purpose is to extend the results obtain in the paper by M. Eleuteri, P. Marcellini, E. Mascolo, Advances in Calculus of Variations (2020).
16:50 — 17:10Raffaella Giova

Regularity results for bounded solutions to obstacle problems

Abstract – I will present some regularity results for bounded solutions of a class of obstacle problems of the form
$$
\min \left\{ \int_ \Omega f(x, Dv) d x \mid v \in \mathcal{K}_\psi (\Omega)\right\}
$$
where $\psi$ is the obstacle, $\mathcal{K}_\psi (\Omega) = \left\{ v \in W_0^{1,p}(\Omega, \mathbb{R}) \mid v \ge \psi \ a.e. \ \text{in} \ \Omega \right\}$ is the class of the admissible functions and the integrand $f(x, Dv)$ satisfies Sobolev regularity with respect to the spatial variable and $p$-standard growth conditions (see [1]) as well as non standard $(p, q)$-growth conditions with respect to the gradient variable (see [2]).

[1] M. Caselli, A. Gentile, R. Giova, Regularity results for solutions to obstacle problems with Sobolev coefficients, J. Differential Equations 269 (2020) 8308–8330
[2] A. Gentile, R. Giova, A. Torricelli, Regularity results for bounded solutions to obstacle problems with non-standard growth conditions, Preprint (2021)
17:10 — 17:30Closure chaired by Sergio Polidoro