Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis (PCA for short) approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper, based on the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms.

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α ²Δu + u(1 − u ²) = 0 in a smooth bounded domain Ω ⊂ R³, with Neumann boundary condition and α > 0 a small parameter. These solutions have the property that as α → 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ∂Ω orthogonally and that is non-degenerate respect to ∂Ω. The authors provide explicit examples of surfaces to which the result applies.

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional and is of Caputo-type, which takes into account “memory”. The precise model is $D_t^\alpha u - div\left( {u - {{\left( { - \Delta } \right)}^{ - \sigma }}u} \right) = f,0 < \sigma < \frac{1}{2}.$ This paper poses the problem over {t ∈ R⁺, x ∈ R ⁿ} with nonnegative initial data u(0, x) ≥ 0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x) have exponential decay at infinity is proved. The main result is Hölder continuity for such weak solutions.

Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon, precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory (see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.

In this paper, the authors prove an analogue of Gibbons’ conjecture for the extended fourth order Allen-Cahn equation in ℝ ^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.

This paper contains a detailed, self contained and more streamlined proof of the l ² decoupling theorem for hypersurfaces from the paper of Bourgain and Demeter in 2015. The authors hope this will serve as a good warm up for the readers interested in understanding the proof of Vinogradov’s mean value theorem from the paper of Bourgain, Demeter and Guth in 2015.

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.

In this expository article, the authors discuss the connection between the study of non-local operators on Euclidean space to the study of fractional GJMS operators in conformal geometry. The emphasis is on the study of a class of fourth order operators and their third order boundary operators. These third order operators are generalizations of the Dirichlet-to-Neumann operator.

This paper offers a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on ℝ ⁿ. The authors use this idea to extend the result to real hyperbolic spaces ℍ ⁿ.

The authors discuss the existence and uniqueness up to isometries of E ⁿ of immersions ϕ: Ω ⊂ R ⁿ → E ⁿ with prescribed metric tensor field (g _ij): Ω → S_> ⁿ, and discuss the continuity of the mapping (g _ij) → ϕ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping (g _ij) → ϕ can be obtained by means of nonlinear Korn inequalities is shown.

The author reviews some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian.

Asymptotic expansions of the voltage potential in terms of the “radius” of a diametrically small (or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.

In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold, σ, are examples of inhomogeneities for which the convergence to the background potential, or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.

The purpose of this paper is to find a “simple” replacement for the background potential, with the following properties: (1) This replacement may be (simply) calculated from the limiting domain Ωσ, the boundary data on the boundary of Ω, and the right-hand side. (2) This replacement depends on the thickness of the inhomogeneity and the conductivity, a, through its boundary conditions on σ. (3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.

The authors prove that flat ground state solutions (i.e. minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions N = 1,2 and they can be stable for N ≥ 3 for suitable values of the involved exponents.