We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta-Ka... more We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the L^1 topology
We consider local minimizers of the functional ∑_i=1^N ∫ (|u_x_i|-δ_i)^p_+ dx+∫ f u dx, where δ_1... more We consider local minimizers of the functional ∑_i=1^N ∫ (|u_x_i|-δ_i)^p_+ dx+∫ f u dx, where δ_1,...,δ_N> 0 and ( · )_+ stands for the positive part. Under suitable assumptions on f, we prove that local minimizers are Lipschitz continuous functions if N=2 and p> 2, or if N> 2 and p> 4.
Abstract. We study a free interface problem of finding the optimal energy configuration for mixtu... more Abstract. We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin
We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>... more We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also provide a similar estimate in the Euclidean case for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.
In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{... more In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{x \in \Omega}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by Champion, De Pascale and Prinari.
We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta-Ka... more We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the L^1 topology
We consider local minimizers of the functional ∑_i=1^N ∫ (|u_x_i|-δ_i)^p_+ dx+∫ f u dx, where δ_1... more We consider local minimizers of the functional ∑_i=1^N ∫ (|u_x_i|-δ_i)^p_+ dx+∫ f u dx, where δ_1,...,δ_N> 0 and ( · )_+ stands for the positive part. Under suitable assumptions on f, we prove that local minimizers are Lipschitz continuous functions if N=2 and p> 2, or if N> 2 and p> 4.
Abstract. We study a free interface problem of finding the optimal energy configuration for mixtu... more Abstract. We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin
We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>... more We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also provide a similar estimate in the Euclidean case for the enlargement with a general convex set. This is equivalent to the stability of the Brunn-Minkowski inequality for the Minkowski sum between a convex set and a generic one.
In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{... more In this paper the existence of an absolute minimizer for a functional \[ F(u,\Omega) = \underset{x \in \Omega}{\text{ess sup}} \, f (x, u(x), Du(x)) \] is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by Champion, De Pascale and Prinari.
In this paper we are concerned with the approximation of functions by single hidden layer neural ... more In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincare inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R...
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