Berardino Sciunzi

Regularity and comparison principles for p-Laplace equations with vanishing source term

We prove some sharp estimates on the summability properties of the second derivatives of solutions to the equation –Δpu = f(x), under suitable assumptions on the source term. As an application, we deduce some strong comparison principles for the p-Laplacian, in the case of vanishing source terms.

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.

A strong comparison principle for the p-laplacian

We consider weak solutions of the differential inequality of p-Laplacian type - Δpu - f(u) ≤-Δpv - f(v) such that u p - 1.

Blow up of solutions of semilinear heat equations in general domains

Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent . Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

Classification of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN

Abstract We provide the classification of the positive solutions to − Δ p u = u p ⁎ − 1 in D 1 , p ( R N ) in the case 2 p N . Since the case 1 p ≤ 2 is already known this provides the complete classification for 1 p N .

A uniqueness result for some singular semilinear elliptic equations

Given

On the moving plane method for nonlocal problems in bounded domains

\Omega

Classification of positive

a bounded open subset of

\mathcal {D}^{1,p}(\R^N)

\mathbb{R}^N

-solutions to the critical

, we consider nonnegative solutions to the singular semilinear elliptic equation