Open Subset Of Rn Research Articles

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: Gϵ∗=sup∫ΩG(u)ϵq(x)dx:∥∇u∥Lp(.)≤ϵ,u=0 on ∂Ω, with the help of Γ-convergence, where G:R→R is upper semicontinuous, non zero in the L1 sense, 0≤G(u)≤c|u|q(.), Ω is a bounded open subset of Rn, n≥3, 1<p(.)<n, and p(.)≤q(.)≤p∗(.). For special choices of G, we can study Bernoulli’s free-boundary and plasma problems in variable exponent Lebesgue spaces.

Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations F ( x , u , D u , D 2 u ) = 0 F(x,u,Du,D^2u)=0 in Ω \Omega , where Ω \Omega is an open subset of R N \mathbb {R}^N , and the validity of the strong maximum principle for F ( x , u , D u , D 2 u ) = f F(x,u,Du,D^2u)=f in Ω \Omega , with f ∈ C ( Ω ) f\in \mathrm {C}(\Omega ) being nonpositive. We obtain geometric characterizations of positivity sets { x ∈ Ω : u ( x ) > 0 } \{x\in \Omega \,:\, u(x)>0\} of nonnegative supersolutions u u and establish the strong maximum principle under some geometric assumption on the set { x ∈ Ω : f ( x ) = 0 } \{x\in \Omega \,:\, f(x)=0\} .

The Hausdorff Measure on n-Dimensional Manifolds in ℝm and n-Dimensional Variations

We extend the notion of the variation Vf([a; b]) of a function f : [a; b] → ℝ to the variation Vf(A) of a continuous map f : G → ℝn, where G is an open subset of ℝn, over a set A ⊂ G of the form A = ∪i ∈ IKi where I is countable and all Ki are compact. Let f : G → ℝm where G ⊂ ℝn with n ≤ m, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < in ≤ m, let fα be the map with coordinate functions $$ {f}_{i_1},\dots, {f}_{i_n} $$. The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝn, and a subset A ⊂ G has the form A = ∪i ∈ IKi where I is countable and all Ki are compact, then $$ {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) $$ where $$ {V}_{f_{\alpha }}(A) $$ is the variation of fα over A and Hn is the n-dimensional Hausdorff measure in ℝm.

Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev space W1,p(Ω), where Ω is an open subset of RN and p∈(1,∞), then u∈W01,p(Ω) if u/d belongs to weak Lp(Ω), where d(x)= dist x,∂Ω. Results of this type are given here for Sobolev spaces with a variable exponent p, under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log-Holder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that u∈W01,p(Ω) if and only if u∈W1,p(Ω) and u/d∈L1(Ω).

Absolutely continuous functions with values in a Banach space

Let Ω be an open subset of Rn, n>1, and let X be a Banach space. We prove that α-absolutely continuous functions f:Ω→X are continuous and differentiable (in some sense) almost everywhere in Ω.

Pointwise bounds and blow-up for systems of semilinear parabolic inequalities and nonlinear heat potential estimates

We study the behavior for t small and positive of C2,1 nonnegative solutions u(x,t) and v(x,t) of the system0≤ut−Δu≤vλ0≤vt−Δv≤uσ in Ω×(0,1), where λ and σ are nonnegative constants and Ω is an open subset of Rn, n≥1. We provide optimal conditions on λ and σ such that solutions of this system satisfy pointwise bounds in compact subsets of Ω as t→0+. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.

Stabilization of the wave equation with Ventcel boundary condition

We consider a damped wave equation on a open subset of Rn or a smooth Riemannian manifold with boundary, with Ventcel boundary conditions, with a linear damping, acting either in the interior or at the boundary. This equation is a model for a vibrating structure with a layer with higher rigidity of thickness δ>0. By means of a proper Carleman estimate for second-order elliptic operators near the boundary, we derive a resolvent estimate for the wave semigroup generator along the imaginary axis, which in turn yields the logarithmic decay rate of the energy. This stabilization result is obtained uniformly in δ.

On some properties of smooth sums of ridge functions

The following problem is studied: If a finite sum of ridge functions defined on an open subset of Rn belongs to some smoothness class, can one represent this sum as a sum of ridge functions (with the same set of directions) each of which belongs to the same smoothness class as the whole sum? It is shown that when the sum contains m terms and there are m − 1 linearly independent directions among m linearly dependent ones, such a representation exists.

Caffarelli–Kohn–Nirenberg type equations of fourth order with the critical exponent and Rellich potential

We study the existence/nonexistence of positive solutions ofΔ2u−μu|x|4=|u|qβ−2u|x|βin Ω, where Ω is a bounded domain and N≥5, qβ=2(N−β)N−4, 0≤β<4 and 0≤μ<(N(N−4)4)2. We prove the nonexistence result when Ω is an open subset of RN, which is star-shaped with respect to the origin. We also study the existence of positive solutions when Ω is a smooth bounded domain with a nontrivial topology and β=0, μ∈(0,μ0), for certain μ0<(N(N−4)4)2 and N≥8. Different behaviors are obtained for Palais–Smale sequences depending on whether β=0 or β>0.

Blow-up for the wave equation with nonlinear source and boundary damping terms

The paper deals with blow-up for the solutions of an evolution problem consisting in a semilinear wave equation posed in a boundedC1,1open subset of ℝn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied iswhere∂Ω=Γ0∪Γ1,Γ0∩Γ1= ∅,σ(Γ0) &gt; 0, 2 &lt;p≤ 2(n− 1)/(n− 2) (whenn≥ 3),m&gt; 1,α∈L∞(Γ1),α≥ 0 andβ≥ 0. The initial data are posed in the energy space.The aim of the paper is to improve previous blow-up results concerning the problem.

Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of higher order with analytic coefficients

We prove that every fundamental solution of an elliptic linear partial differential operator with analytic coefficients in Ω open subset of Rn and simple complex characteristics can be holomorphically continued at least locally as a multi-valued analytic function x⟶E(x,y) in Cn up to the complex bicharacteristic conoid. This holomorphic extension is ramified around the bicharacteristic conoid and belongs to the Nilsson class.

Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets ofRn.

Piecewise linear approximation of smooth functions of two variables

Given a piecewise linear (PL) function p defined on an open subset of Rn, one may construct by elementary means a unique polyhedron with multiplicities D(p) in the cotangent bundle Rn×Rn⁎ representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth function f(x,y) may be approximated by a sequence p1,p2,… of PL functions such that the areas of the D(pi) are locally dominated by the area of the graph of df times a universal constant.

A RELATIVE ISOPERIMETRIC INEQUALITY

We prove that, if Ω is an open subset of ℝN with finite measure, there exists a hyperplane H through 0 such that the measure of Ω ∩ H is less than the measure of B ∩ H, where B is the open ball with center 0 having the same measure as Ω. An application is given to the optimal Poincaré inequality on BV(Ω).

Generalization by Optimization in Menger Probabilistic Normed Spaces with Application on Portfolio in Insurance

The purpose of this paper is to show that the category of normed spaces can be embedded in the category of Menger probabilistic normed spaces, and that C( Ω ) is probabilistic normable, whereas it is not normable in the classical case, when Ω is an open subset of Rn. So, the spectrum of the category of Menger probabilistic normed spaces is broader than the category of classical normed spaces. Therefore, it can be a meaningful replacement in some model of security markets. As our model is suitably generalized, that fact can help us to adapt and improve within natural problems of finance, especially, in portfolio optimization of insurance.

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

In (0,T)×Ω, Ω open subset of ℝ n , n≥2, we consider a parabolic operator P=∂ t −∇ x δ(t,x)∇ x , where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderon projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hormander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.

An approach via symmetrization methods to nonlinear elliptic problems with a lower order term

In this paper we consider a class of nonlinear elliptic problems of the type $$ \left\{ \begin{gathered} - div(a(x,\nabla u)) - div(\Phi (x,u)) = fin\Omega \hfill \\ u = 0on\partial \Omega , \hfill \\ \end{gathered} \right. $$ where Ω is a bounded open subset of R N , N ≥ 2, f is a L 1 (Ω) function or a Radon measure with bounded total variation. We fix some structural conditions on a and Φ to prove uniqueness results when f ∈ L 1 (Ω).

A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators

In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form $$ \mathcal{L} u:=\sum\limits_{j=1}^m X_j^2u+X_0u-\partial_tu=0 $$ where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and \(X_0 - \partial_t\) are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0,t0) ∈ Ω we give a geometric sufficient condition on the compact sets \(K\subseteq {\Omega}\) for which the Harnack inequality $$ \sup\limits_{K}u\le C_K\, u(x_0,t_0) $$ holds for all non-negative solutions u to the equation \(\mathcal{L} u=0\). We also compare our result with an abstract Harnack inequality from potential theory.

Universal Taylor series on open subsets of Rn

Universal Taylor series on open subsets of R^<i>n</i>

A fixed point theorem for a class of Sobolev maps

We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of R n .