Schedule for: 17w5116 - Mostly Maximum Principle

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I explain an approach, based on generalized Mather measures, to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such PDEs, which are natural extensions of Mather measures, originally due to J. Mather. Using the viscosity Mather measures, one can show that the whole family of solutions $v^\lambda$ of the discounted problem, with the discount factor $\lambda$, converges to a solution of the ergodic problem as $\lambda$ goes to 0. This is based on joint work with Hiroyoshi Mitake (Hiroshima University) and Hung V. Tran (University of Wisconsin, Madison).

We will discuss the existence, nonexistence and concentration of solutions of fully non linear elliptic equations of the type : $$-F(x,D^2u)=|u|^{p-1}u$$ with homogeneous boundary conditions in bounded domains. In particular, similarities and differences with respect to the semilinear case will be emphasized. This is a joint research with I.Birindelli, F.Leoni and G.Galise.

The regularity theory for fully nonlinear PDEsEpodes was inaugurated in the turn of 70s to the 80s, with the works of Krylov and Safonov. In the span of a decade, a number of important advances took place in the field; e.g., the Evans-Krylov theory and Caffarelli's estimates in Sobolev spaces. In face of those breakthroughs, a question on the possibility of a general theory for that class of problems naturally arose. Such a question was set in the negative only recently; in a series of papers, Nadirashvili and Vladut produced a number of counterexamples unveiling the subtleties of this theory. In this talk, we examine the regularity of the solutions to elliptic and parabolic fully nonlinear equations, in Sobolev spaces. We make use of a geometric-tangential approach, which enables us to work under asymptotic assumptions of the operator governing the problem. We also put forward a number of new estimates and applications, consequential to our results.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, I will present the recent results obtained in collaboration with L. Silvestre (Chicago) about Hölder continuity of solutions of the Boltzmann equation without cut-off under the condition that mass, energy and density are locally bounded and that mass is bounded away from zero (non-vacuum condition). Such a regularity result is a consequence of a weak Harnack inequality for a general kinetic integro-differential equation for kernels satisfying sufficiently mild conditions to be applicable to the Boltzmann equation. Such a weak Harnack inequality is obtained by applying elliptic regularity techniques of De Giorgi type.

Maximum principles at infinity (or ``almost maximum principles") are a powerful tool to investigate the geometry of Riemannian manifolds. In this talk I will focus on the the Ekeland, the Omori-Yau principles and their weak versions, in the sense of Pigola-Rigoli-Setti. These last have probabilistic interpretations in terms of stochastic and martingale completeness, that is, the non-explosion of (respectively) the Brownian motion and each martingale on $M$. After an overview to show the usefulness of these principles in geometry, I will move to discuss necessary and sufficient conditions for their validity. In particular, I will focus on an underlying duality that allows to discover new relations between them. Indeed, duality holds for a broad class of fully-nonlinear operators of geometric interest. Our methods use the approach to nonlinear PDEs pioneered by Krylov ('95) and Harvey-Lawson ('09 - ). This is based on joint works with B. Bianchini, M. Rigoli, P. Pucci and L. F. Pessoa.

In this work, in collaboration with Emmanuel Chasseigne (Tours) and Thi Tuyen Nguyen (Rennes), we establish a priori Lipschitz estimates for unbounded solutions of viscous Hamilton-Jacobi equations in presence of a Ornstein-Uhlenbeck drift. The first type of equations we consider are local. The Ornstein-Uhlenbeck drift is associated with a general diffusion operator. This part is a generalization of an earlier work of Fujita, Ishii & Loreti (2006). The second type of equations we deal with are nonlocal. The Ornstein-Uhlenbeck term is associated with a integro-differential operator of Fractional Laplacian type. In both case, we obtain some local Lipschitz estimates which are independent of the L^\infty norm of the solution (which is supposed to have at most an exponential growth). These results can be applied to prove the large time behavior of the solutions.

The aim of the talk is to discuss some recent results on structure conditions which allow solutions of fully nonlinear elliptic equations, to be continued - or not - through a singular set or throughout the whole space.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In a series of papers, F. Hamel, N. Nadirashvili and E. Russ deal with the isoperimetric problem for eigenvalues of second order elliptic operators. With respect to the classical Faber-Krahn inequality, they consider an additional drift term, which acts as a control under L infinity constraint. They solve the problem and derive the precise asymptotic of the principal eigenvalue as the norm of the drift tends to infinity. As a preliminary step, they consider the optimization problem in a fixed domain. This leads to a nonlinear eigenvalue problem. They conjecture that, in such case, the maximal points of the optimal principal eigenfunction approach the points with maximal distance from the boundary, and that its gradient aligns with the gradient of the distance function. We present a proof of the first conjecture and give a partial result concerning the second one. These results have been obtained in collaboration with F. Hamel and E. Russ.

We consider nonlinear parabolic equations of the form $$u_t= \Delta u+V(x)f(u),$$ where $V\ge 0$ is a potential and $f$ is a nonlinearity which can be either of blow-up or of quenching type, typically $$f(u)=(1+u)^p\quad (p>1) \quad\hbox{ or }\quad f(u)=(1-u)^{-p}\quad (p>0).$$ We are interested in the localization of finite time singularities. A main question is: can one rule out the occurence of singularities at zeros of the potential ? More generally, can one show that singularities concentrate at points where $V$ is large in a certain sense ? We will present positive and negative results, which indicate that the issue is delicate. The techniques involve combinations of various kinds of maximum principle arguments, as well as Liouville type theorems. Joint works in collaboration with Jong-Shenq Guo and with Carlos Esteve.

In this talk I will deal with some recent results, obtained with D. De Silva and S. Salsa, about C^{1,\gamma} regularity and higher regularity of free boundaries of solutions of some non-homogeneous elliptic two phase problems.

We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with a nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,\gamma}$ with $\gamma \in (0,1]$. We consider both bounded and exterior domains. This is a joint work with Pier Domenico Lamberti.

We will show two versions of Aleksandrov-Bakelman-Pucci maximum principles for integro-PDE of Hamilton-Jacobi-Bellman type whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical ABP maximum principles for the elliptic and parabolic PDE and an argument using an iteration procedure involving solutions of Pucci extremal equations. This is a joint work with Chenchen Mou.

In this talk we will present an overview of some recent results on α-harmonic maps which are horizontal with respect to a given plane distribution.

In this talk, we discuss a quantitative inhomogeneous version for the Hopf-Oleinik Lemma for Fully nonlinear and Quasilinear PDEs. This allows us to obtain several interior and up to the boundary regularity results for these equations and related free boundary problems. In this talk, we focus on the regularity of the normal mapping for supersolutions as an extension of the gradient estimates obtained by Caffarelli-Kohn-Nirenberg-Spruck in the mid 80's. We will survey other results if time permits.

We discuss the validity of the maximum principle below the principal eigenvalue for viscosity solutions of the Dirichlet problem in bounded domains $$ {\cal P}^-_{k}(D^2u)+H(x,\nabla u)+\mu u=0\quad\mbox{in}\;\Omega,\quad u=0\quad\mbox{on}\;\partial \Omega, $$ where the higher order term is given by the truncated Laplacian $ {\cal P}^-_{k}(D^2u)=\sum_{i=1}^k\lambda_i(D^2u), $ being $\lambda_i(D^2u)$ being the ordered eigenvalues of the Hessian. Some very unusual phenomena due to the degeneracy of the operator will be emphasized by means of explicit counterexamples. We shall present moreover global Lipschitz regularity results and boundary estimates in the context of convex domains for $k=1$, leading to the existence of a principal eigenfunction. Some open question will be raised. This is joint work with I. Birindelli and H. Ishii.

We study global solutions $u:{\mathbb R}^3\to{\mathbb R}^2$ of the Ginzburg-Landau equation $-\Delta u=(1-|u|^2)u$ which are local minimizers in the sense of De Giorgi. We prove that a local minimizer satisfying the condition $\liminf_{R\to\infty}\frac{E(u;B_R)}{R\ln R}<2\pi$ must be constant. The main tool is a new sharp $\eta$-ellipticity result for minimizers in dimension three that might be of independent interest. This is a joint work with Etienne Sandier (Universit\'e Paris-Est).

We consider elliptic equations on the entire space of 2 or more dimensions. Using center manifold and KAM theorems, we show the existence of solutions which are quasiperiodic in one variable and decay in all the other variables.

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework. This is a joint work work with Serena Dipierro and Enrico Valdinoci.

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, $$\begin{array}{c} u_t = \Delta u + |\nabla u|^2 u \quad &\text{in } \Omega\times(0,T)\\ u = \vp \quad &\text{on } \pp\Omega\times(0,T)\\ u(\cdot,0) = u_0 \quad & \text{in } \Omega \end{array} $$ where $\Omega$ is a bounded, smooth domain in $\R^2$ and $u: \Omega\times(0,T)\to S^2$, $u_0:\bar\Omega \to S^2$, smooth, $\vp= u_0\big|_{\pp\Omega}$. Given any points $q_1,\ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We analyze stability of this phenomenon if $k=1$. This is joint work with Juan D\'avila and Juncheng Wei.

I will discuss some questions regarding the conjecture of De Giorgi on the Allen-Cahn equation and which remain still open. The talk will be mainly concerned with the saddle-shaped solution in all of $\R^{2m}$. A remarkable open problem is to establish that this solution is a minimizer in high dimensions ---more precisely, this is believed to be true for $2m \geq 8$. The saddle-shaped solution is odd with respect to the Simons cone and exists in all even dimensions. I will explain results of the author and collaborators which establish: the uniqueness of the saddle-shaped in every even dimension $2m \geq 2$, its instability in dimensions 2, 4, and 6, and its stability for $2m \geq 14$. I will also describe results of Pacard and Wei, and a very recent one by Liu, Wang, and Wei, which construct a family of global minimizers in $\R^8$. If this family includes the saddle-shaped solution is still unknown.

The first Neumann eigenfunction is a constant, while the first Dirichlet eigenfunction is known to be log-concave, by a result of Brascamp and Lieb. It seems reasonable to conjecture that the first Robin eigenfunction should also be log-concave. However in recent work we have shown that this cannot be true. This is joint work with Ben Andrews and Daniel Hauer.

In this talk we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\R^N_+=\{x=(x',x_N)\in \R^N:\ x_N>0\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $\rho:=\sup_{\mathbb{R}^N}u$ verifying $f(\rho)=0$, then $u$ is necessarily one-dimensional.

In this talk, we develop a connection between an ordinary differential equation that is cubic in the unknown function, and the Kuramoto-Sivashinsky equation, a partial differential equation that occupies a prominent position in describing some physical processes in motion of turbulence and other unstable process systems. We convert the problem into an equivalent integral equation by using the Abel transformation. By means of the Lie symmetry reduction method and the Maximum Principle, we show that there exist nontrivial bounded solitary wave solutions under certain parametric conditions. Numerical simulations of wave phenomena are illustrated, which provide us rich dynamical information and are in agreement with our theoretical analysis.

We consider in the whole plane the Hamiltonian coupling of Schrödinger equations where the nonlinearities have critical or even supercritical growth in the sense of Moser. In the critical case, we prove that the (nonempty) set of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to singularly perturbed systems. Namely, in presence of an external potential which is bounded away from zero, we prove the existence of minimal energy solutions which concentrate around the closest local minima of the potential with some precise asymptotic rate. In the supercritical case we prove the existence of higher (though finite) energy solutions in a suitable Lorentz space framework.

For fully nonlinear k-Hessian operators on bounded strictly (k-1)-convex domains of Euclidian space, a characterization of the principal eigenvalue associated to a k-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of elliptic branches in the sense of Krylov [TAMS’95] that correspond to selecting k-convex functions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property coming from the establishment of a global Hölder continuity property for the approximating equations. This is joint work with Isabeau Birindelli.

We study the existence of nonnegative supersolutions of the nonlinear elliptic problem $-\Delta u + |\nabla u|^q = \lambda u^p$ in the half-space $\R^N_+$, where $N\ge 2$, $q>1$, $p>0$ and $\la>0$. We obtain Liouville theorems for positive, bounded supersolutions, depending on the exponents $q$ and $p$, the dimension $N$, and, in some critical cases, also on the parameter $\la>0$.

In this talk we consider the problem of characterizing spheres in C^2 by the fact they have constant Levi curvature. We discuss a rigidity result of Jellett-type for a suitable class of real hypersurfaces. The strong maximum principle for a particular subelliptic operator on the hypersurface plays a crucial role in this approach. As a main application, we provide an Aleksandrov-type result for starshaped domains with circular symmetries. This is a joint work with V. Martino.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.