Mean curvature properties for pp-Laplace phase transitions

This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of p-Laplacian type and a double well potential h(0) with suitable growth conditions. We prove that level sets of solutions of Delta(p)u=h(0)'(u) possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature

Denoising and enhancement of mammographic images under the assumption of heteroscedastic additive noise by an optimal subband thresholding

Mammographic images suffer from low contrast and signal dependent noise, and a very small size of tumoral signs is not easily detected, especially for an early diagnosis of breast cancer. In this context, many methods proposed in literature fail for lack of generality. In particular, too weak assumptions on the noise model, e.g., stationary normal additive noise, and an inaccurate choice of the wavelet family that is applied, can lead to an information loss, noise emphasizing, unacceptable enhancement results, or in turn an unwanted distortion of the original image aspect. In this paper, we consider an optimal wavelet thresholding, in the context of Discrete Dyadic Wavelet Transforms, by directly relating all the parameters involved in both denoising and contrast enhancement to signal dependent noise variance (estimated by a robust algorithm) and to the size of cancer signs. Moreover, by performing a reconstruction from a zero-approximation in conjunction with a Gaussian smoothing filter, we are able to extract the background and the foreground of the image separately, as to compute suitable contrast improvement indexes. The whole procedure will be tested on high resolution X-ray mammographic images and compared with other techniques. Anyway, the visual assessment of the results by an expert radiologist will be also considered as a subjective evaluation

Monotonicty of solutions of fully nonlinear uniformly elliptic equation in the half-plane

In this paper we study the monotonicity of positive (or nonnegative) viscosity solutions to uniformly elliptic equation

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f locally Lipschitz continuous, and prove some regularity results for weak C-1() solutions. In particular when f (s) > 0 for s > 0 we prove summability properties of 1/\Du\, and Sobolev's and Poincare type inequalities in weighted Sobolev spaces with weight \Du\(m-2). The point of view of considering \Du\(m-2) as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f (s) > 0 for s > 0 and m > 2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1 < m < 2. (C) 2004 Elsevier Inc. All rights reserved

Qualitative properties of solutions of m-Laplace systems

We prove regularity results for the solutions of the equation -Delta(m)u = h(x), such as summability properties of the second derivatives and summability properties of 1/vertical bar Du vertical bar. Analogous results were recently proved by the authors for the equation -Delta(m)u = f (u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem {-Delta(m1)(u) = f (v) u > 0 in Omega, u = 0 on theta Omega {-Delta(m2)(v) = g(u) v > 0 in Omega, v = 0 on theta Omega and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Z(u) equivalent to {x is an element of Omega vertical bar Du(x) = 0} and Z(v) equivalent to {x is an element of Omega vertical bar Dv(x) = 0}. In particular we prove that in convex and symmetric domains we have Z(u) = {0} - Z(v), assuming that 0 is the center of symmetry

Spectral theory for linearized p-Laplace equations

We continue and completely set up the spectral theory initiated in Castorina et al. [D. Castorina, P. Esposito, B. Sciunzi, Degenerate elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 34 (2009), 279–306] for the linearized operator arising from Δ_p u+f(u)=0. We establish existence and variational characterization of all the eigenvalues, and by a weak Harnack inequality we deduce Hölder continuity for the corresponding eigenfunctions, this regularity being sharp. The Morse index of a positive solution can be now defined in the classical way, and we will illustrate some qualitative consequences one should expect to deduce from such information. In particular, we show that zero Morse index (or more generally, nondegenerate) solutions on the annulus are radial

A strong comparison principle for the p-laplacian

We consider weak solutions of the differential inequality of p-Laplacian type -(p)u - f(u) <= - Delta(p)v - f(v) such that u <= v on a smooth bounded domain in RN and either u or v is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that u < v on the boundary of the domain we prove that u < v, and assuming that u equivalent to v equivalent to 0 on the boundary of the domain we prove u < v unless u equivalent to v. The novelty is that the nonlinearity f is allowed to change sign. In particular, the result holds for the model nonlinearity f(s) = s(q) - lambda s(p-1) with q > p - 1

On a Poincar&#233; type formula for solutions of singular and degenerate elliptic equations

We provide a geometric Poincar\ue9 type formula for stable solutions of -\u394p(u) = f(u). From this, we derive a symmetry result in the plane. This work is a refinement of previous results obtained by the authors under further integrability and regularity assumptions