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Advances in Mathematics, ISSN 0001-8708, 04/2018, Volume 328, pp. 40 - 81
In this paper we prove the local Lipschitz regularity of the minimizers of the two-phase Bernoulli type free boundary problem arising from the minimization of... 
Partial regularity | Free boundary regularity | p-Laplace | Monotonicity formula | Two phase
Journal Article
Communications in Mathematical Physics, ISSN 0010-3616, 1/2017, Volume 349, Issue 2, pp. 583 - 626
We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another... 
Quantum Physics | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | DISLOCATION DYNAMICS | EXISTENCE | LAGRANGIAN SYSTEMS | CRYSTALS | LIMIT | DIFFUSION | MODEL | PHYSICS, MATHEMATICAL | REVERSIBLE HAMILTONIAN SYSTEM | Mathematics - Analysis of PDEs
Journal Article
Communications in Mathematical Physics, ISSN 0010-3616, 1/2015, Volume 333, Issue 2, pp. 1061 - 1105
We consider an evolution equation arising in the Peierls–Nabarro model for crystal dislocation. We study the evolution of such a dislocation function and show... 
Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | SLOW MOTION | PEIERLS-NABARRO MODEL | PARTICLE-SYSTEMS | LIMIT | PHYSICS, MATHEMATICAL | EQUATION | HOMOGENIZATION
Journal Article
Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 08/2013, Volume 33, Issue 8, pp. 3473 - 3496
We obtain some Poincare type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable... 
Elliptic systems | Stable solutions | Phase separa- tion | Poincaré-type inequality | Monotone solutions | MATHEMATICS | monotone solutions | MATHEMATICS, APPLIED | phase separation | stable solutions | Poincare-type inequality
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 07/2018, Volume 43, Issue 7, pp. 1073 - 1101
We continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a... 
monotonicity formula | p-Laplace | Two phase | free boundary regularit | Primary: 35R35 | 35J92 | MATHEMATICS | FLAT FREE-BOUNDARIES | MATHEMATICS, APPLIED | LIPSCHITZ REGULARITY | EQUATIONS | MINIMUM PROBLEM | DOMAINS | Free boundaries
Journal Article
Mathematische Annalen, ISSN 0025-5831, 12/2017, Volume 369, Issue 3, pp. 1283 - 1326
We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we... 
Mathematics, general | Mathematics | EXISTENCE | MATHEMATICS | LAPLACIANS | SYMMETRY | NONLINEAR EQUATIONS | MU-TRANSMISSION | NONNEGATIVE SOLUTIONS | UNIQUENESS | Mathematics - Analysis of PDEs
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 8/2016, Volume 55, Issue 4, pp. 1 - 29
We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s... 
35A15 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35B33 | 35R11 | Mathematics | 35B40 | MATHEMATICS | MATHEMATICS, APPLIED | SYMMETRY | REGULARITY
Journal Article
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, ISSN 0218-2025, 12/2019, Volume 29, Issue 14, pp. 2585 - 2636
We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such... 
EXISTENCE | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | crystal dislocation | PATTERNS | LAYER SOLUTIONS | DISPERSAL | Peierls-Nabarro model | PHASE | EVOLUTION | REGULARITY | fractional Laplacian | ORBITS | Heteroclinic solutions | nonlocal equations
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 01/2013, Volume 254, Issue 1, pp. 30 - 66
We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain Ω⊂Rn whose boundary has an... 
Local inversion | Finite-dimensional reductions | Singularly perturbed elliptic problems | EXISTENCE | SYSTEM | MATHEMATICS | STATES | NONLINEAR SCHRODINGER-EQUATIONS | SPIKE-LAYER SOLUTIONS | LEAST-ENERGY SOLUTIONS | SEMILINEAR NEUMANN PROBLEM | MULTIPEAK SOLUTIONS
Journal Article
Applied Mathematics & Optimization, ISSN 0095-4616, 12/2019, Volume 80, Issue 3, pp. 679 - 713
Journal Article
The Journal of Geometric Analysis, ISSN 1050-6926, 4/2019, Volume 29, Issue 2, pp. 1428 - 1455
We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function... 
60G22 | Approximation | 35A35 | Mathematics | s -caloric functions | 34A08 | Abstract Harmonic Analysis | Fourier Analysis | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | 35R11 | Differential Geometry | Dynamical Systems and Ergodic Theory | Density properties | s-caloric functions | MATHEMATICS
Journal Article
Journal of Statistical Physics, ISSN 0022-4715, 06/2017, Volume 167, Issue 6, pp. 1401 - 1451
Journal Article
Journal of Evolution Equations, ISSN 1424-3199, 6/2019, Volume 19, Issue 2, pp. 435 - 462
We consider an evolution equation whose time-diffusion is of fractional type, and we provide decay estimates in time for the $$L^s$$ L s -norm of the solutions... 
58D25 | Parabolic equations | Decay of solutions in time with respect to Lebesgue norms | Fractional diffusion | Analysis | 26A33 | 35K90 | Mathematics | 34A08 | 47J35 | MATHEMATICS | MATHEMATICS, APPLIED
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 2/2018, Volume 57, Issue 1, pp. 1 - 21
We consider bounded solutions of the nonlocal Allen–Cahn equation $$\begin{aligned} (-\Delta )^s u=u-u^3\qquad { \text{ in } }\mathbb {R}^3, \end{aligned}$$... 
82B26 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 35R11 | 35B65 | Mathematics | FRACTIONAL LAPLACIANS | MATHEMATICS | MATHEMATICS, APPLIED | MINIMAL-SURFACES | NONLINEAR EQUATIONS | REGULARITY | DIFFUSION | ELLIPTIC-EQUATIONS
Journal Article
Annali di Matematica Pura ed Applicata (1923 -), ISSN 0373-3114, 8/2017, Volume 196, Issue 4, pp. 1327 - 1344
Boggio’s formula in balls is known for integer-polyharmonic Dirichlet problems and for fractional Dirichlet problems with fractional parameter less than 1. We... 
Mathematics, general | Mathematics | 35J40
Journal Article
Mathematische Annalen, ISSN 0025-5831, 10/2019, Volume 375, Issue 1, pp. 687 - 736
We discuss fattening phenomenon for the evolution of sets according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures... 
35R11 | Mathematics, general | 35D40 | Mathematics | 53C44 | 58E12 | MATHEMATICS | EVOLUTION | MOTION | MEAN-CURVATURE | SETS
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 8/2016, Volume 55, Issue 4, pp. 1 - 25
In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in... 
53A10 | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | 49Q05 | 35R11 | Mathematics | MATHEMATICS | MATHEMATICS, APPLIED | LIMITING ARGUMENTS
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 12/2014, Volume 39, Issue 12, pp. 2351 - 2387
We consider the equation where L s is an integro-differential operator of order 2s, with s ∈ (0, 1), W is a periodic potential, and σ ϵ is a small external... 
35B05 | Oscillation and regularity results | 49N60 | Nonlocal Peierls-Nabarro model | 35Q99 | 35D30 | 35B40 | Fractional Laplacian | 35J25 | Dislocation dynamics | 35G25 | MATHEMATICS | MATHEMATICS, APPLIED | PARTICLE-SYSTEMS | EQUATION | HOMOGENIZATION | Partial differential equations | Stresses | Mathematical analysis | Slip | Images | Crystals | Evolution | Mathematical models | Dislocations
Journal Article
International Mathematics Research Notices, ISSN 1073-7928, 04/2018
Journal Article
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