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We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class of bounded Lipschitz domains in $\mathbb{R}^n$. We also prove that any corresponding positive eigenfunction is log-concave if the domain is convex., Comment: 21 pages

In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to $0^+$ and to $+\infty$. Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meet the convex envelope of the fixed boundary., Comment: 18 pages

We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restrict ourselves to starshaped rings in a warped product of an open interval and the unit sphere. We also extend the result by replacing the Laplacian with the $q$-Laplacian., Comment: 16 pages. Comments welcome!

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.

In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Pr\'ekopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions.

We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge $\Theta$ for which we introduce a suitable notion of Legendre transform $\Theta^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $\Theta^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.

We deal with Monge-Amp\`ere type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on $H$, on the gradient image of the domain. The Wulff shape symmetry associated with $H$ of the solutions is established.

$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely: (1) we discover hot-concavity, which is the strongest $F$-concavity preserved by the Dirichlet heat flow; (2) we show that log-concavity is the weakest $F$-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$; (3) we prove that if $F$-concavity does not coincide with log-concavity and it is not stronger than log-concavity and $n\ge 2$, then there exists an $F$-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for $F$-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy--Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic $p$-Laplace equation., Comment: 45pages. Comments are welcome

We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational approach, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions., Comment: 42 pages

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative power concave initial datum may result in losing immediately any reminiscence of concavity). Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the Dirichlet heat flow, we see that log-concavity is indeed the only power concavity preserved by the Dirichlet heat flow., Comment: The authors withdrew this preprint because its conclusion is now contained as a corollary in arXiv:2207.13449 (see Corollary 1.7)

We introduce a notion of $F$-concavity which largely generalizes the usual concavity. By the use of the notions of closedness under positive scalar multiplication and closedness under positive exponentiation we characterize power concavity and power log-concavity among nontrivial $F$-concavities, respectively. In particular, we have a characterization of log-concavity as the only $F$-concavity which is closed both under positive scalar multiplication and positive exponentiation. Furthermore, we discuss the strongest $F$-concavity preserved by the Dirichlet heat flow, characterizing log-concavity also in this connection., Comment: 21pages. Comments are welcome!

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $\Gamma$ be the heat kernel on ${\bf H}^N$. Then $\Gamma(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$., Comment: 24 pages. Comments are welcome!

This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results apply to the Pucci operator, the normalized $q$-Laplacians with $1

In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow., Comment: We removed one result, Theorem 3.2 of the old version, due to a gap in the proof

In this paper we establish the starshapedness of the level sets of the capacitary potential of a large class of fully-nonlinear equations for condensers in Carnot groups, once a natural notion of starshapedness has been introduced. Our main result is Theorem 1.2 below., Comment: Accepted for publication in the Journal of the London Mathematical Society

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $\Omega \subset \mathbb{R}^N$, with $1

In the present work we study solutions of the problem $-(-\Delta)^{\alpha/2}u = f(x,u)$ in $D_0\setminus \overline{D}_1$, with exterior conditions $u = 0$ in $R^N \setminus D_0$ and $u = 1$ in $\overline{D}_1$, where $D_1, D_0 \subset R^N$ are open sets such that $\overline{D}_1 \subset D_0$, $\alpha \in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x} \in \overline{D}_1$, then the same occurs for every superlevel set of $u$.

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs., Comment: to appear in GAFA Seminar Notes

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in $\mathbb{R}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).

We strengthen, in two different ways, the so called Borell-Brascamp- Lieb inequality in the class of power concave functions with compact support. As examples of applications we obtain two quantitative versions of the Brunn- Minkowski inequality and of the Urysohn inequality for torsional rigidity., Comment: We modified section 7

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

We introduce a method to compare solutions of different equations in different domains. As a consequence, we define a new kind of rearrangement which applies to solution of fully nonlinear equations $F(x,u,Du,D^2u)=0$, not necessarily in divergence form, in convex domains and we obtain Talenti's type results for this kind of rearrangement.

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) & \mbox{in}\quad\Omega\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial \Omega\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quad\Omega, \end{array} \right. \end{equation} where $\Omega$ is a bounded convex domain in ${\bf R}^n$ and $f$ is a nonnegative continuous function in $\Omega\times(0,\infty)\times{\bf R}\times{\bf R}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\bar{\Omega}\times[0,\infty)$.

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.

Let $p\in(1,n)$. If $\Omega$ is a convex domain in $\rn$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.

We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Amp\`ere equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations., Comment: 20 pages, 5 figures