101,949 research outputs found
On the Dimension of the Singular Set in Optimization Problems with Measure Constraint
No full textWe prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraint. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program recently published by G. Buttazzo, F. P. Maiale, D. Mazzoleni, G. Tortone and B. Velichkov [Regularity of the optimal sets for a class of integral shape functionals, arxiv 2212.09118 (2022)] to this class of domain variation
Sunto delle lezioni di economia politica / G. G. Reymond
No full textSunto delle lezioni di economia politica / G. G. Reymond
Torino : Speirani e Tortone, 1859
XIV, 323 p. (1 op.; posseduto solo 48 p.
Combustion of natural gas on premixed catalysed metal fibre burners aged and poisoned by S-compounds
No full text
Regularity of the Optimal Sets for a Class of Integral Shape Functionals
No full textWe prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain Ω is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on Ω. The main feature of these functionals is that the minimality of a domain Ω cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions j(u,x)=-g(x)u+Q(x), where u is the solution of the PDE -Δu=f with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of Ω and then we use the stability of Ω with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose ∂Ω into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of ∂Ω we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove C∞ regularity of the regular part of the free boundary when the data are smooth
Epsilon-regularity for the solutions of a free boundary system
No full textThis paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v, anda domain st; with u and v being both positive in st, vanishing simultaneously on 8st, and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on 8st. Precisely, we consider solu-tions u, v E C(B1) of- u = f and -v = g in Omega = {u > 0} = {v > 0},partial derivative u partial derivative v/partial derivative n partial derivative n on partial derivative Omega boolean AND n B-1.Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions ,/uv and 1/2 (u + v). Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C-1,C-alpha regularity
Liouville theorems and optimal regularity in elliptic equations
No full textThe objective of this paper is to establish a connection between the problem
of optimal regularity among solutions to elliptic PDEs with measurable
coefficients and the Liouville property at infinity. Initially, we address the
two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity
formula, enabling the proof of optimal regularity and the Liouville property
for multiphase problems. In higher dimensions, we delve into the role of
monotonicity formulas in characterizing optimal regularity. By employing a
hole-filling technique, we present a distinct "almost-monotonicity" formula
that implies Hlder regularity of solutions. Finally, we explore the
interplay between the least growth at infinity and the exponent of regularity
by combining blow-up and -convergence arguments
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