### Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

## Talks

Wednesday, July 14, 16:00 ~ 16:30 UTC-3

## On the Monge-Ampere system

### Marta Lewicka

#### University of Pittsburgh, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Monge-Ampere equation $\det\nabla^2 u =f$ posed on a $N=2$ dimensional domain, has a natural weak formulation that appears as the constraint condition in the $\Gamma$-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: $curl^2 (\nabla v\otimes \nabla v) = -2f$ and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in $C^0(\omega)$, at the regularity $C^{1,\alpha}$ for any $\alpha<1/7$. $\\$

Does a similar result hold in higher dimensions $N>2$? Indeed it does, but one has to replace the Monge-Ampere equation by a {\em “Monge-Ampere system”}, altering $curl^2$ to the corresponding operator whose kernel consists of the symmetrised gradients of $N$-dimensional displacement fields. We will show how this Monge-Ampere system arises from the prescribed Riemannian curvature problem by matched asymptotic expansions, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d, and prove that its flexibility at $C^{1,\alpha}$ for any $\alpha<1/(N^2+N+1)$.

Wednesday, July 14, 16:30 ~ 17:00 UTC-3

## On the Principal Frequency of the Anisotropic $p$-Laplacian

### Marian Bocea

#### National Science Foundation, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the monotonicity (with respect to $p$) of the principal frequency of the anisotropic $p$-Laplacian on bounded, convex domains with smooth boundary. As a consequence of our main result, we obtain a new variational characterization for the principal frequency on domains having a sufficiently small inradius. The talk is based on joint work with Denisa Stancu-Dumitru ("Politehnica" University of Bucharest, Romania) and Mihai Mihailescu (University of Craiova, Romania).

Wednesday, July 14, 17:00 ~ 17:30 UTC-3

## Herglotz-Nevanlinna functions and fluid permeability tensors

### Miao-jung Yvonne Ou

#### University of Delaware, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Herglotz-Nevanlinna functions are closely related to self-adjoint operators and have been used for finding useful analytical properties of physical quantities defined from elliptic equations. In this talk, we will show the link of Herglotz-Nevanlinna functions and the permeability of fluid mixtures and how they are linked to the permeability of porous media.

Joint work with This is joint work with Bi Chuan (NIA/NIH) and Shangyou Zhang(U. of Delaware)..

Wednesday, July 14, 17:30 ~ 18:00 UTC-3

## Simple motion of stretch-limited elastic strings

### Casey Rodriguez

#### University of North Carolina at Chapel Hill, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

Wednesday, July 14, 18:00 ~ 18:30 UTC-3

## Short-time existence for the network flow

### Mariel Saez

#### Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard-Reitich [5], Mantegazza-Novaga- Tortorelli and eventually, in the most general case of irregular networks by Ilmanen- Neves-Schulze. Although the present paper proves a result similar to the one in the paper by Ilmanen- Neves-Schulze, the method here provides substantially more detailed information about how an irregular network ‘resolves’ into a regular one. Either approach relies on the existence of self-similar expanding solutions found in previous work.

Joint work with Joint with Jorge Lira, Rafe Mazzeo and Alessandra Pluda.

Wednesday, July 14, 18:30 ~ 19:00 UTC-3

## Non-local equation with critical growth on compact Riemannian manifolds.

### Nicolas Saintier

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

On a compact Riemannian manifold $(M,g)$ of dimension $n$ we consider a non-local equation of the form \[ \mathcal{L}_K u + hu = f |u|^{2^*-2} u \] where the operator $\mathcal{L}_K u$ is given in weak form by \[ (\mathcal{L}_K u,v) = \iint_{M\times M} (u(x)-u(y))(v(x)-v(y))K(x,y;g)\,dv_g(x)dv_g(x). \] The kernel $K(x,y;g)$ essentially behaves like $1/d_g(x,y)^{n+2s}$, $s\in (0,1)$, and satisfies some useful properties when the metric is blown-up at a point. The exponent $2^*:=2n/(n-2s)$ is critical from the point of view of Sobolev embedding. This kind of non-local equation on $\mathbb{R}^n$ has been the subject of an intense research activity in the past years.

We first study the associated functional spaces $\widetilde{W}^{s,2}(M) $ of the functions $u\in L^2(M)$ such that $ [u]_{s,2}<\infty$ where \[ u_{s,2}^2:= \int_{M\times M} \frac{|u(x)-u(y)|^2}{d_g(x,y)^{n+2s}}\,dv_g(x)dv_g(y). \] We then establish an optimal Sobolev inquality of the form \[ \Big( \int_M |u|^{2^*} \,dv_g\Big)^\frac{2}{2^*} \le (A(n,s,2)+\varepsilon) \iint_{M\times M} |u(x)-u(y)|^2 K(x,y;g)\,dv_g(x)dv_g(y) + C_\varepsilon \int_M u^2\,dv_g \] valid for any $\varepsilon>0$. Here $A$ is the least constant such that this inequality holds for any $u$, and is given by \[ A(n,s,2)^{-1} = \inf_u \frac{ \displaystyle \iint_{\mathbb{R}^n\times \mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dxdy}{\|u\|_{2^*}^2}. \] This inequlity then allows to state in a standard way a sufficient existence condition.

Joint work with Carolina Rey (Univ. Buenos Aires).

Wednesday, July 14, 19:00 ~ 19:30 UTC-3

## Spectral Stability in the nonlinear Dirac equation with Soler-type nonlinearity in dimension $1$.

### Hanne Van Den Bosch

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies $\omega$ in $(0,m)$, where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction. The results that I will discuss concern simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

Joint work with Danko Aldunate (Pontificia Universidad Católica de Chile), Julien Ricaud (LMU Munich) and Edgardo Stockmeyer (Pontificia Universidad Católica de Chile).

Wednesday, July 14, 19:30 ~ 20:00 UTC-3

## Homogenization of non-dilute suspension of a viscous fluid with magnetic particles

### Thuyen Dang

#### University of Houston, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk presents a rigorous homogenization result of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. The coefficients of the corresponding partial differential equations are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for the formal asymptotic analysis of Levy and Sanchez-Palencia for particulate steady-state Stokes flows.

Joint work with Yuliya Gorb (National Science Foundation, USA) and Silvia Jimenez Bolanos (Colgate University, USA).

Thursday, July 15, 16:00 ~ 16:30 UTC-3

## Quantitative Estimates of Frequency Band-gaps in Photonic Crystals

### Robert Lipton

#### Louisiana State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

A frequency band-gap inside a photonic crystal is an interval of frequencies where there is no wave propagation through the crystal. The size and location of a frequency band-gap depends on the geometry of the periodic array of scatterers. We obtain explicit and rigorously quantitive estimates on band-gap location and size in terms of the geometry of the scatters. Examples are provided for different scatterer configurations and shapes.

These estimates are given in terms of formulas that explicitly depend on the eigenvalues of the Neumann-Poincare operator defined on the boundary of the scatters, the Dirichlet spectrum of the scatter, the scatter configuration, and the contrast in dielectric properties between scatterer and connected phase. The methodology is operator theoretic and uses a new analytic representation of the time harmonic wave equation together with analytic perturbation theory about high contrast. These methods deliver rigorously convergent power series in the contrast from which estimates can be made.

Joint work with Robert Viator (Swarthmore College, Pennsylvania).

Thursday, July 15, 16:30 ~ 17:00 UTC-3

## A Sharp Divergence Theory with Non-Tangential Traces

### Irina Mitrea

#### Temple University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Integration by Parts Formula, which is equivalent with the Divergence Theorem, is one of the most basic tools in Analysis. Originating in the works of Gauss, Ostrogradsky, and Stokes, the search for an optimal version of this fundamental result continues through this day and these efforts have been the driving force in shaping up entire subbranches of mathematics, like Geometric Measure Theory.

In this talk I will review some of these developments (starting from elementary considerations to more sophisticated versions) and I will discuss recents result regarding a sharp divergence theorem with non-tangential traces. This is joint work with D. Mitrea and M. Mitrea.

Thursday, July 15, 17:00 ~ 17:30 UTC-3

## Old and New Perspectives on Effective Equations

### Matthew Rosenzweig

#### Massachusetts Institute of Technology, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we consider nonlinear-Schr\"odinger-type equations as partial differentiation equations (PDEs) arising as effective descriptions of systems of finitely many interacting bosons. We approach this topic from two perspectives. The \emph{old} perspective consists of proving quantitative convergence in an appropriate function space of solutions to the finite problem to a solution of an effective, limiting PDE, as the number of particles tends to infinity. The \emph{new} perspective consists of proving qualitative convergence of geometric structure, such as the properties of being an integrable and Hamiltonian system. Through these two complementary perspectives, focusing on both quantitative and qualitative convergence, we gain a deeper understanding of how field theories, both classical and quantum, may be deformed to a new field theory, and of how this deformation may be reversed.

Joint work with Dana Mendelson (University of Chicago), Andrea Nahmod (University of Massachusetts, Amherst), Natasa Pavlovic (University of Texas at Austin) and Gigliola Staffilani (Massachusetts Institute of Technology).

Thursday, July 15, 17:30 ~ 18:00 UTC-3

## Asymptotic results in magnetic Orlicz-Sobolev spaces

### Ariel Salort

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a non-local and non-standard growth version of the so-called magnetic Laplacian. This operator is known as the magnetic fractional $g-$Laplacian $(-\Delta_g^A)^s$ and it is defined as the gradient of the non-local energy functional \[ I_{s,G}^A(u) := \iint_{\mathbb{R}^n\times\mathbb{R}^n} \left(G\left(|\Re(D_s^A u(x,y))|\right) + G\left(|\Im(D_s^A u(x,y))|\right)\right)\, \frac{dxdy}{|x-y|^n}, \] where $G$ is a Young function and $D_s^A u(x,y)$ is the magnetic Holder quotient of order $s\in(0,1)$ defined as \[ D_s^A u(x,y) := \frac{u(x)-e^{i(x-y) A\left(\frac{x+y}{2}\right)} u(y)}{|x-y|^s}. \] Here $A:\mathbb{R}^n \to \mathbb{R}^n$ is a vector potential and $i$ denotes the imaginary unit.

We introduce briefly the notion of magnetic spaces in the fractional Orlicz-Sobolev setting and we study suitable Maz'ya-Shaposhnikova and Bourgain-Brezis-Mironescu formulas in modular form.

Thursday, July 15, 18:00 ~ 18:30 UTC-3

## Bloch waves in 3-dimensional high-contrast photonic crystals

### Robert Viator

#### Swarthmore College, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

We investigate the Bloch eigenvalues of a 3-dimensional high-contrast photonic crystal. The Bloch eigenvalues (for a fixed quasi-momentum) can be expanded in a power series in the material contrast parameter $k$ about $k=\infty$. We achieve this power series, together with a radius of convergence, by decomposing an appropriate vector-valued Sobolev space into three mutually orthogonal subspaces which are curl-free in certain subdomains of the period cell. We will also identify the limit spectral problem as contrast becomes large, and (time permitting) we will describe a class of crystal geometries which permit the power series structure of Bloch eigenvalues described above.

Joint work with Silvia Jimenez (Colgate University) and Robert Lipton (Louisiana State University).

Thursday, July 15, 18:30 ~ 19:00 UTC-3

## Axisymmetric penny-shaped fracture problem at nanoscale

### Anna Zemlyanova

#### Kansas State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann-Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution of this system can be obtained by expanding the unknown functions into the Fourier-Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behavior of the material system. In particular, presence of surface energy leads to the size-dependency of the solutions and increased toughness of the material.

Thursday, July 15, 19:00 ~ 19:30 UTC-3

## Obstacle type problems with gradient constrains.

### Héctor Chang-Lara

#### Centro de Investigación en Matemáticas - Guanajuato, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the regularity of solutions of equations of the form \[ \min(-\Delta u,|Du|−1)=0 \] These arise when computing the optimal strategy for a zero-sum game involving Brownian and constant speed dynamics determined by the agents. In collaboration with Pimentel we established the optimal Lipschitz regularity and the free boundary condition on the interface. Whenever the first agent fixes its strategy the equation becomes a transmission problem between the Laplace and eikonal equation. In different collaboration with Arellano we analyze the well posedness of the problem and the convergence of a numerical method.

Joint work with Edgard Pimentel (Pontifícia Universidade Católica do Rio de Janeiro, Brasil) and Arturo Arellano (Centro de Investigación en Matemáticas, Guanajuato, México).

Thursday, July 15, 19:30 ~ 20:00 UTC-3

## Uncertainty principles and interpolation formulae in analysis and PDE

### João Pedro Gonçalves Ramos

#### ETH Zürich, Switzerland - This email address is being protected from spambots. You need JavaScript enabled to view it.

The famous Heisenberg uncertainty principles predicts, in one of its versions, that a functions $f$ and its Fourier transform $\widehat{f}$ cannot both be too concentrated in space, otherwise $f \equiv 0.$ This intriguing principle has many classical counterparts of similar flavour, such as the Hardy uncertainty principle and the Amrein-Berthier theorem on annihilating pairs.

In recent years, however, several breakthrough results have shed new light onto problems in the uncertainty realm. In particular, we mention first, in the realm of PDEs, a series of papers by Escauriaza-Kenig-Ponce-Vega, which generalised the Hardy uncertainty to a more general Schrödinger equation setting in its sharp form. Secondly, in the purely analytics realm, we mention the Radchenko-Viazovska interpolation formula, which provides one with an explicit way to recover an even function $f \in \mathcal{S}(\mathbb{R})$ given its values $f(\sqrt{n}), \widehat{f}(\sqrt{n}), n \ge 0.$

In this talk, we will go through some of these uncertainty results, from the most classical to some of the most recent, mentioning possible directions, open problems and conjectures in this area. In particular, we shall emphasise more ideas than technical proofs, making this talk an invitation to contribute in the subject.

Joint work with Mateus Sousa (BCAM, Bilbao, Spain), Christoph Kehle (ETH Zürich, Switzerland) and Martin Stoller (EPFL, Switzerland).

Wednesday, July 21, 16:00 ~ 16:30 UTC-3

## Uncertainty Quantification and Well-Posedness of Damage Mechanics

### Petr Plechac

#### University of Delaware, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Many important materials, including glasses and ceramics, are of the linearly elastic/brittle variety, obeying a classical Hookean law until an applied load reaches a yield stress, at which point they break into some number of fragments. Predicting the behavior of such materials, through modelling and simulation, is of obvious value to a variety of applications. In this work, we add randomness in material properties into the problem, modelling them via random fields within a damage mechanics framework. By identifying key regularizations we establish the well-posedness of a model derived from PDEs of elastodynamics. We then apply tools from uncertainty quantification to investigate the robustness of predictions in the model under materials properties described by random fields.

Joint work with Gideon Simpson (Drexel University, USA) and Jerome Troy (University of Delaware, USA).

Wednesday, July 21, 16:30 ~ 17:00 UTC-3

## Elastic bodies with fractures: duality and homogenization

### Bogdan Vernescu

#### Worcester Polytechnic Institute, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the stationary elasticity problem for a solid with fractures, and review, in the framework of Legendre-Fenchel duality, three equivalent formulations for the problem in terms of displacement, stress, and strain. For periodically distributed fractures, we prove a homogenization result using Mosco-convergence in the $L^2$- topology.

Joint work with Riuji Sato (Worcester Polytechnic Institute).

Wednesday, July 21, 17:00 ~ 17:30 UTC-3

## Efficient Numerical Treatment of High-Contrast Composite Materials

### Yuliya Gorb

#### National Science Foundation, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk concerns a robust numerical treatment of an elliptic PDE with high contrast coefficients. We introduce a procedure by which a discrete system obtained from a linear finite element discretization of the given continuum problem is converted into an equivalent linear system of the saddle point type. Preconditioners for solving the derived saddle point problem that are robust with respect to the contrast parameter and the discretization scale are proposed and justified.

Joint work with Yuri Kuznetsov (University of Houston).

Wednesday, July 21, 17:30 ~ 18:00 UTC-3

## Multiscale preconditioners for linear elastic topology optimization

### Galvis Juan

#### Universidad Nacional de Colombia, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

We present a novel fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process requires the numerical solution of large high-contrast linear elastic problems with features spanning several length scales. The size of the discretized problems and the lack of clear separation between the scales, as well as the high-contrast, imposes severe challenges on the standard preconditioning techniques. We propose novel methods for the high-contrast elasticity equation with performance independent of the high-contrast and the multi-scale structure of the elasticity problem. The solvers are based on two-levels domain decomposition techniques with a carefully constructed coarse level to deal with the high-contrast and multi-scale nature of the problem. The new methods inherit the advantages of domain decomposition techniques, such as easy parallelization and scalability. The presented numerical experiments demonstrate the excellent performance of the proposed methods. This talk is based on https://doi.org/10.1016/j.cam.2020.113366.

Joint work with Boyan Lazarov (Lawrence Livermore National Laboratory, Livermore, CA, US), Sintya Serrano (Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE) and Miguel Zambrano (Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE).

Wednesday, July 21, 18:00 ~ 18:30 UTC-3

## Effect of non-linear lower order terms in quasilinear equations involving the $p(x)$-Laplacian

### Analía Silva

#### UNSL-IMASL, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we study the existence of $W_0^{1, p(x)}$-solutions to the following boundary value problem involving the $p(x)$-Laplacian operator:

$$ \left\lbrace \begin{array}{l} -\Delta_{p(x)}u+|\nabla u|^{q(x)}=\lambda g(x)u^{\eta(x)}+f(x), \quad in \Omega, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq 0, \quad in \Omega\\ \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad on \partial\Omega.\\ \end{array} \right. $$under appropriate ranges on the variable exponents. We give assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $\eta$ under which the above problem has a solution for all $\lambda > 0$.

Joint work with Pablo Ochoa (UNCUYO).

Wednesday, July 21, 18:30 ~ 19:00 UTC-3

## Multiple solutions for the fractional nonlinear Schrödinger equation

### Salomón Alarcón

#### Universidad Técnica Federico Santa María, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the equation \begin{equation*}\label{ppp} \varepsilon^{2s}(-\Delta)^s u +V(x)u -f(u)=0\quad\mbox{in }\mathbb{R}^N, \tag{$P$} \end{equation*} where $s\in (0,1)$, $p\in \big(1,\frac{N+2s}{N-2s}\big)$, $N> 2s$, $f(u)=|u|^{p-1}u$, $V\in L^{\infty}(\mathbb{R}^N)$ is such that $\inf_{\mathbb{R}^N}V>0$ and $\varepsilon>0$ is small. Via a reduction method, we construct positive and sign-changing solutions concentrating at a saddle point of $V$ and sign-changing solutions of ($P$) concentrating at a local minimum point of $V$ as $\varepsilon\rightarrow0$. To guarantee the existence we cannot neglect the interaction between peaks. The proof of all our results is based in a max-min scheme which uses topological degree, extending some known results from the nonlinear case involving the Laplacian operator.

References

[1] T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1423-1451.

[2] T. D'Aprile and D. Ruiz, Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems, Math. Z. 268 (2011), 605-634.

[3] J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858-892.

[4] X. Kang, J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations 5 (2000), 899-928.

Joint work with Antonella Ritorto (Universiteit Utrecht, Germany) and Analía Silva (Universidad Nacional de San Luis, Argentina).

Wednesday, July 21, 19:00 ~ 19:30 UTC-3

## A viscosity solution approach to regularity properties of the optimal value function

### Pablo Ochoa

#### CONICET-Universidad Nacional de Cuyo, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this paper, we analyze the optimal value function $v$ associated to a general parametric optimization problem via the theory of viscosity solutions. The novelty is that we obtain regularity properties of $v$ by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of $v$. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get further differentiability properties of $v$ in Euclidean spaces.

Joint work with Virginia N. Vera de Serio (Universidad Nacional de Cuyo, Argentina).