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We study the regularity of solutions of the following semilinear problem(Formula presented.)where B1 is the unit ball in ℝn, 0 < q < 1 and λ± satisfy a Hölder continuity condition. Our main results concern local regularity analysis of solutions and their nodal set {u = 0}. The desired regularity is C[κ],κ−[κ] for κ = 2/(1 − q) and we divide the singular points in two classes. The first class contains the points where at least one of the derivatives of order less than κ is nonzero, the second class which is named (Formula presented.), is the set of points where all the derivatives of order less than κ exist and vanish. We prove that (Formula presented.) when κ is not an integer. Moreover,... 

We study the regularity of solutions of the following semilinear problemΔu=−λ+(x)(u+)q+λ−(x)(u−)qinB1,where B1 is the unit ball in ℝn, 0 < q < 1 and λ± satisfy a Hölder continuity condition. Our main results concern local regularity analysis of solutions and their nodal set {u = 0}. The desired regularity is C[κ],κ−[κ] for κ = 2/(1 − q) and we divide the singular points in two classes. The first class contains the points where at least one of the derivatives of order less than κ is nonzero, the second class which is named Sκ, is the set of points where all the derivatives of order less than κ exist and vanish. We prove that Sκ= ∅ when κ is not an integer. Moreover, with an example we show... 

The singular sources method is given to detect the shape of a thin infinitely cylindrical obstacle from a knowledge of the TM-polarized scattered electromagnetic field in large distance. The basic idea is based on the singular behaviour of the scattered field of the incident point source on the cross-section of the cylinder. We assume that the scatterer is a perfect conductor which is possibly coated by a material and investigate two models with different boundary conditions. Also we give a uniqueness proof for the shape reconstruction. Copyright © 2006 John Wiley & Sons, Ltd  

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation ut - (a(x)ux)x - λ/x βu = 0, (t,x) ∈ (0, T) × (0,1), where the diffusion coefficient a(·) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(·). Under some conditions on the function a(·) and parameters β, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality  

We study the null controllability properties of some degenerate/singular parabolic equations in a bounded interval of ℝ. For this reason we derive a new Carleman estimate whose proof is based on Hardy inequalities  

The purpose of this paper is to address the question of well-posedness and spectral controllability of the wave equation perturbed by potential on networks which may contain unbounded potentials in the external edges. It has been shown before that in the absence of any potential, there exists an optimal time T∗ (which turns out to be simply twice the sum of all length of the strings of the network) that describes the spectral controllability of the system. We will show that this holds in our case too, i.e., the potentials have no effect on the value of the optimal time T∗. The proof is based on the famous Beurling-Malliavin’s Theorem on the completeness interval of real exponentials and on a... 

We study the semilinear problem Δu=λ+(x)(u+)q−1−λ−(x)(u−)q−1inB1, from a regularity point of view for solutions and the free boundary ∂{±u>0}. Here B1 is the unit ball, 1