1,720,993 research outputs found
A note on the monotonicity formula of Caffarelli-Jerison-Kenig
Full text linkThe aim of this note is to prove the monotonicity formula of Caffarelli-Jerison-Kenig for functions, which are not necessarily continuous. We also give a detailed proof of the multiphase version of the monotonicity formula in any dimension
On the logarithmic epiperimetric inequality for the obstacle problem
Full text linkWe give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities
An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case
Full text linkUsing a direct approach, we prove a two-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc
A shape optimal control problem with changing sign data
No full textIn this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal domain exists. We also deduce some necessary conditions of optimality for the optimal domain. The results are applied to show the existence of an optimal domain in the case where the cost functional is completely identified, while the right-hand side in the state equation is only known up to a probability P in the space L2(D)
Multiphase shape optimization problems
Full text linkThis paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min (Formula presented.) where D ⊆ Rd is a given bounded open set, |Ωi| is the Lebesgue measure of Ωi, and m is a positive constant. For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., Fi = λki
Existence and regularity of minimizers for some spectral functionals with perimeter constraint
Full text linkIn this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ Rd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ N and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition. © 2013 Springer Science+Business Media New York
Regularity of the free boundary for the vectorial Bernoulli problem
Full text linkWe study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D ⊂ Rd, Λ > 0, and φ[symbol]i ∈ H1/2.(∂D), we deal with min [ ∫D [pipe]∇υi [pipe]2 +Λ [υi ≠ 0 ] [pipe]: υi + φ[symbol] i on ∂ D]. We prove that, for any optimal vector U = (u1,..., uk), the free boundary ∂ (∪ki=1 [ui ≠ 0] [n-ary intersection] D is made of a regular part, which is relatively open and locally the graph of a C∞ function, a (one-phase) singular part, of Hausdorff dimension at most d-d, for a d ∈ [5, 6, 7], and by a set of branching (two-phase) points, which is relatively closed and of finite Hd-1 measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem
Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem
Full text linkWe answer a question left open in [4] and [3], by proving that the blow-up of minimizers u of the lower dimensional obstacle problem is unique at generic point of the free boundary
A logarithmic epiperimetric inequality for the obstacle problem
Full text linkWe study the regularity of the regular and of the singular set of the obstacle problem in any dimension. Our approach is related to the epiperimetric inequality of Weiss (Invent Math 138:23–50, Wei99a), which works at regular points and provides an alternative to the methods previously introduced by Caffarelli (Acta Math 139:155–184, Caf77). In his paper, Weiss uses a contradiction argument for the regular set and he asks the question if such epiperimetric inequality can be proved in a direct way (namely, exhibiting explicit competitors), which would have significant implications on the regularity of the free boundary in dimension d > 2. We answer positively the question of Weiss, proving at regular points the epiperimetric inequality in a direct way, and more significantly we introduce a new tool, which we call logarithmic epiperimetric inequality. It allows to study the regularity of the whole singular set and yields an explicit logarithmic modulus of continuity on the C1 regularity, thus improving previous results of Caffarelli and Monneau and providing a fully alternative method. It is the first instance in the literature (even in the context of minimal surfaces) of an epiperimetric inequality of logarithmic type and the first instance in which the epiperimetric inequality for singular points has a direct proof. Our logarithmic epiperimetric inequality at singular points has a quite general nature and will be applied to provide similar results in different contexts, for instance for the thin obstacle problem
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