Academic literature on the topic 'Nonlocal isoperimetric problems'

AbstractWe show a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under volume and convexity constraints.We prove existence of minimizers, and we describe their shape as the volume tends to zero or to infinity.

AbstractIn this article, we study scaling laws for simplified multi-well nucleation problems without gauge invariances which are motivated by models for shape-memory alloys. Seeking to explore the role of the order of lamination on the energy scaling for nucleation processes, we provide scaling laws for various model problems in two and three dimensions. In particular, we discuss (optimal) scaling results in the volume and the singular perturbation parameter for settings in which the surrounding parent phase is in the first-, the second- and the third-order lamination convex hull of the wells of the nucleating phase. Furthermore, we provide a corresponding result for the setting of an infinite order laminate which arises in the context of the Tartar square. In particular, our results provide isoperimetric estimates in situations in which strong nonlocal anisotropies are present.

The thesis is divided into two parts. The first part is mainly concerned with regularity issues for integro-differential (or nonlocal) elliptic and parabolic equations. In the same way that densities of particles with Brownian motion solve second order elliptic or parabolic equations, densities of particles with Lévy diffusion satisfy these more general nonlocal equations. In this context, fully nonlinear nonlocal equations arise in Stochastic control problems or differential games. The typical example of elliptic nonlocal operator is the fractional Laplacian, which is the only translation, rotation and scaling invariant nonlocal elliptic operator. There many classical regularity results for the fractional Laplacian ---whose ``inverse'' is the Riesz potential. For instance, the explicit Poisson kernel for a ball is an ``old'' result, as well as the linear solvability theory in L^p spaces. However, very little was known on boundary regularity for these problems. A main topic of this thesis is the study of this boundary regularity, which is qualitatively very different from that for second order equations. We stablish a new boundary regularity theory for fully nonlinear (and linear) elliptic integro-differential equations. Our proofs require a combination of original techniques and appropriate versions of classical ones for second order equations (such as Krylov's method). We also obtain new interior regularity results for fully nonlinear parabolic nonlocal equation with rough kernels. To do it, we develop a blow up and compactness method for viscosity solutions to fully nonlinear equations that allows us to prove regularity from Liouville type theorems.This method is a main contribution of the thesis. The new boundary regularity results mentioned above are crucially used in the proof of another main result of the thesis: the Pohozaev identity for the fractional Laplacian. This identity is has a flavor of integration by parts formula for the fractional Laplacian, with the important novely there appears a local boundary term (this was unusual with nolocal equations). In the second part of the thesis we give two instances of interaction between isoperimetry and Partial Differential Equations. In the first one we use the Alexandrov-Bakelman-Pucci method for elliptic PDE to obtain new sharp isoperimetric inequalities in cones with densities by generalizing a proof of the classical isoperimetric inequality due to Cabré. Our new results contain as particular cases the classical Wulff inequality and the isoperimetric inequality in cones of Lions and Pacella. In the second instance we use the isoperimetric inequality and the classical Pohozaev identity to establish a radial symmetry result for second order reaction-diffusion equations. The novelty here is to include discontinuous nonlinearities. For this, we extend a two-dimensional argument of P.-L. Lions from 1981 to obtain now results in higher dimensions
La tesi està dividida en dues parts. La primera part es centra principalment en questions de regularitat per equacions integro - iferencials (o no locals) el·líptiques i parbòliques. De la mateixa manera que les densitats de partícules amb un moviment Brownià resolen equacions el·líptiques o parbòliques de segon ordre, les densitats de partícules amb una difusió de tipus Lévy resolen aquestes equacions no locals més generals. En aquest context, les equacions completament no lineals sorgeixen de problemes de control estocàstic o "differential games''. L'exemple típic d'operador el·liptic no local és el laplacià fraccionari, el qual és l'únic d'aquests operadors que és invariant per translacions, rotacions, i reescalament. Hi ha molts resultats clàssics de regularitat per el laplacià fraccionari --- "l'invers'' del qual és el potencial de Riesz. Per exemple, el nucli de Poisson (explícit) per la bola és un resultat "vell'', així com la teoria de resolubilitat en espais L^p. No obstant això, se sabia ben poc sobre la regularitat a la vora per a aquests problemes. Un tema principal d'aquesta tesi és l'estudi d'aquesta regularitat a la vora, que és qualitativament molt diferent de la de les equacions de segon ordre . A la tesi s'estableix una nova teoria regularitat a la vora per completament no lineals ( i lineals ) equacions integro - diferencials el·líptiques . Les nostres demostracions requeixen una combinació de tècniques originals i versions apropiades de les clàssiques equacions de segon ordre ( com ara el mètode de Krylov ). També obtenim nous resultats de regularitat interior per equacions parabòliques no locals completament no lineals i amb "rough kernels''. A tal efecte, desenvolupem un mètode de blow-up i compacitat per a equacions completament no lineals que en permet provar regularitat a partir de teoremes de tipus Liouville. Aquest mètode és una contribució principal de la tesi. Els nous resultats de regularitat a la vora esmentats anteriorment són essencials en la prova d'un altre resultat principal de la tesi: la identitat Pohozaev per al Laplacià fraccionari. Aquesta identitat recorda a una fórmula d'integració per parts, però amb el Laplacià fraccionari. La novetat important és que apareix un terme de vora locals (això era inusual amb equacions no locals) . A la segona part de la tesi que donem dos exemples d'interacció entre isoperimetria i Equacions en Derivades Parcials. En el primer, s'utilitza el mètode d'Alexandrov- Bakelman-Pucci per a EDP el·líptiques a fi d'obtenir noves desigualtats isoperimètriques en cons convexos amb densitats, generalitzant una prova de la desigualtat isoperimètric clàssica de X. Cabré. Els nostres nous resultats contenen com a casos particularsla desigualtat clàssica de Wulff i la desigualtat isoperimètrica en cons de Lions-Pacella. En el segon exemple s'utilitza la desigualtat isoperimètrica i la identitat Pohozaev clàssica per establir un resultat de simetria radial per equacions de reacció-difusió de segon ordre. La novetat en aquest cas és que s'inclouen no-linealitats discontínues. Per a provar aquest resultat, estenem un argument en dues dimensions de P.-L. Lions de 1981 i podem obtenir ara resultass en dimensions superiors.

This thesis is concerned with various problems arising in the study of nonlinear elliptic PDE. It is divided into three parts. In the first part we consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. Our mathematical framework is that of multiple time scale systems and singular perturbations. We are concerned with the asymptotic behaviour of a logarithmic functional of the process, which we study by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We provide some financial applications, namely we prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity. In the second part we are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton-Jacobi equations related to jump processes in general (enough smooth) domains. We consider a nonlocal diffusive term of censored type of order less than 1 and Hamiltonian both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron's method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data. In the last part we present some stability results for a class of integral inequalities, the Borell-Brascamp-Lieb inequality and we strengthen, in two different ways, these inequalities in the class of power concave functions. Then we present some applications to prove analogous quantitative results for certain type of isoperimetric inequalities satisfied by a wide class of variational functionals that can be written in terms of the solution of a suitable elliptic boundary value problem. As a toy model, we consider the torsional rigidity and prove quantitative results for its Brunn-Minkowski inequality and for its consequent (Urysohn type) isoperimetric inequality.
Questa tesi si occupa di vari problemi che sorgono nello studio di equazioni alle derivate parziali ellittiche e paraboliche. La tesi è divisa in tre parti. Nella prima parte studiamo il comportamento per tempi brevi di sistemi dinamici a volatilità stocastica che evolve in una scala temporale più veloce.Ci occupiamo di perturbazioni singolari di sistemi a scala temporale multipla. Il nostro primo obiettivo è lo studio del comportamento asintotico di un funzionale logaritmico del processo stocastico, attraverso i metodi della teoria dell' omogeneizzazione e delle perturbazioni singolari per equazioni alle derivate parziali completamente non lineari. Individuiamo tre regimi a seconda della velocità con cui la volatilità oscilla rispetto alla lunghezza dell'orizzonte temporale. Inoltre forniamo alcune applicazioni finanziarie, in particolare proviamo un principio di grandi deviazioni in ogni regime e lo applichiamo per derivare una stima asintotica dei prezzi di opzioni vicino alla maturità e una formula asintotica per la volatilità di Black-Scholes implicita. Nella seconda parte studiamo la buona definizione di problemi al contorno di tipo Neumann, in domini generali (sufficientemente regolari), per equazioni tipo Hamilton-Jacobi con termini non locali che derivano da processi discontinui a salti. Consideriamo un termine diffusivo non locale di tipo censored, di ordine strettamente minore di 1, e un' Hamiltoniana, sia in forma coerciva sia di tipo Bellman non necessariamente coerciva, la cui crescita nel gradiente la rende il termine principale nell'equazione. Dimostriamo un principio di confronto per sotto e sopra soluzioni limitate (in senso di viscosità) con condizioni al contorno generalizzate, e di conseguenza tramite il metodo di Perron otteniamo l'esistenza e l'unicità di soluzioni continue. Diamo alcune applicazioni nel caso evolutivo, dimostrando la convergenza per tempi grandi della soluzione del problema evolutivo alla soluzione del problema stazionario associato, supponendo opportune ipotesi sui dati. Nell'ultima parte presentiamo alcuni risultati di stabilità per una classe di diseguaglianze integrali, le disuguaglianze Borrell-Brascamp-Lieb e rafforziamo, in due modi diversi, queste disuguaglianze nella classe di funzioni a potenza concava. Come applicazione di questo risultato, presentiamo analoghi risultati quantitativi per alcuni tipi di disuguaglianze isoperimetriche soddisfatte da un'ampia classe di funzionali variazionali che possono essere scritti in termini della soluzione di un opportuno problema al contorno ellittico. Come modello giocattolo, consideriamo la rigidità torsionale e dimostriamo risultati quantitativi per la sua disuguaglianza Brunn-Minkowski e per la sua conseguente disuguaglianza isoperimetrica di tipo Urysohn.