Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi$ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.Comment: 34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math. Ser. (N.Y.), vol. 12, Springer, New York 201

Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a byproduct of our technique we are able to answer positively to a conjecture of E.B.Davies.Comment: 40 page

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in \bx, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.Comment: 61 page

Simultaneous occurrence of cerebellar medulloblastoma and pituitary adenoma: A case report

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On the heat kernel of a class of fourth order operators in two dimensions: Sharp Gaussian estimates and short time asymptotics

We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with L∞ coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this work is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case. © 201

Differential immunohistochemical expression of CD44s, E-cadherin and β-catenin among hyperplastic and neoplastic lesions of the prostate gland

Introduction: CD44s, E-cadherin and β-catenin are cell adhesion molecules (CAMs) and appear to influence organ development, inflammation, cancer invasion and metastasis. We studied the expression of these CAMs in prostate cancer (PCa), high-grade prostatic intraepithelial neoplasia (HGPIN) and nodular adenomatous hyperplasia (NH). Materials and Methods: 135 paraffin blocks of radical prostatectomy specimens were assessed. CAMs were determined by immunohistochemistry. All sections included PCa, HGPIN and NH. The expression was semiquantitatively evaluated in three scores (1+, 2+, 3+). The markers' immunopositivity was statistically investigated with Gleason score and TNM stage. Results and Conclusions: CD44s had score 3+ in 41.5, 46.7 and 37.8% of areas with NH, HGPIN and PCa, respectively. E-cadherin immunostaining was highly detected in 71.1, 78.5 and 63.0% of NH, HGPIN and PCa areas while β-catenin score 3+ was exclusively membranous in 80.7% of NH and nuclear/cytoplasmic in 70.4 and 48.9% of HGPIN and PCa areas. No marker related to the Gleason score (p = 0.352). CD44s and E-cadherin expression was inversely associated with TNM stage (p = 0.021 and p = 0.042, respectively); no such association was observed for β-catenin (p = 0.556). The decreased expression of CD44s and E-cadherin is probably associated with the invasive potential of PCa. The β-catenin staining pattern in neoplastic lesions, either preinvasive or invasive, differs from that in non-neoplastic prostate lesions. Copyright © 2012 S. Karger AG, Basel