Ph.D. Thesis
Inserted: 4 nov 2019
Last Updated: 26 mar 2020
Year: 2019
Abstract:
Taking up a variational viewpoint, we present some nonlocal-to-local asymptotic results for various kinds of integral functionals. The content of the thesis comprises the contributions first appeared in some research papers in collaboration with J. Berendsen, A. Cesaroni, A. Chambolle, and M. Novaga.
After an initial summary of the basic technical tools, the first original result is discussed in Chapter 2. It is motivated by a work by J. Bourgain, H. Brezis, and P. Mironescu, who proved that the $L^p$-norm of the gradient of a Sobolev function can be recovered as a suitable limit of iterated integrals involving the difference quotients of the function. A. Ponce later showed that the relation also holds in the sense of $\Gamma$-convergence. Loosely speaking, we take into account the rate of this convergence and we establish the $\Gamma$-converge of the rate functionals to a second order limit w.r.t. the $H^1_{\mathrm{loc}}(\mathbb{R}^d)$-convergence.
Next, from Chapter 3 on, we move to a geometric context and we consider the nonlocal perimeters associated with a positive kernel $K$, which we allow to be singular in the origin. Qualitatively, these functionals express a weighted interaction between a given set and its complement. More broadly, we study a total-variation-type nonlocal functional $J_K(\,\cdot\,;\Omega)$, where $\Omega\subset \mathbb{R}^d$ is a measurable set. We establish existence of minimisers of such energy under prescribed boundary conditions, and we prove a criterion for minimality based on calibrations. Due to the nonlocal nature of the problem at stake, the definition of calibration has to be properly chosen. As an application of the criterion, we prove that halfspaces are the unique minimisers of $J_K$ in a ball subject to their own boundary conditions.
A second nonlocal-to-local $\Gamma$-convergence result is discussed in Chapter 4. We rescale the kernel $K$ so that, when the scaling parameter approaches $0$, the family of rescaled functions tends to the Dirac delta in $0$. If $K$ has small tails at infinity, we manage to show that the nonlocal total variations associated with the rescaled kernels $\Gamma$-converge w.r.t. the $L^1_{\mathrm{loc}}(\mathbb{R}^d)$-convergence to a local, anisotropic total variation.
Lastly, we consider the nonlocal curvature functional associated with $K$, which is the geometric $L^2$-first variation of the nonlocal perimeter. In the same asymptotic regime as above, we retrieve a local, anisotropic mean curvature functional as the limit of rescaled nonlocal curvatures. In particular, the limit is uniform for sets whose boundary is compact and smooth. As a consequence, we establish the locally uniform convergence of the viscosity solutions of the rescaled nonlocal geometric flows to the viscosity solution of the anisotropic mean curvature motion. This is obtained by combining a compactness argument and a set-theoretic approach that relies on the theory of De Giorgi's barriers for evolution equations.
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