The mixed problem in L^p for some two-dimensional Lipschitz domains

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p

An Isoperimetric Inequality for Fundamental Tones of Free Plates

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given $\tau>0$, the free plate eigenvalues $\omega$ and eigenfunctions $u$ are determined by the equation $\Delta\Delta u-\tau\Delta u = \omega u$ together with certain natural boundary conditions. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient, which contains a Hessian squared term $|D^2u|^2$. We adapt Weinberger's method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions. These solutions are a linear combination of Bessel and modified Bessel functions.Comment: PhD thesis. Papers are in preparatio

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω ) or H˜s−2(Ω ) , 12<s<32, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.EPSR

Central blockade of oxytocin receptors during mid-late gestation reduces amplitude of slow afterhyperpolarization in supraoptic oxytocin neurons

The neurohypophysial hormone oxytocin (OT), synthesized in magnocellular paraventricular (PVN) and supraoptic (SON) nuclei, is well known for its effects in lactation. Our previous studies showed that central OT receptor (OTR) binding is increased during gestation and that blockade of central OTRs, specifically during mid-late gestation, causes a delay in OT release during suckling and reduces weight gain in pups, suggesting decreased milk delivery. In the present study, we tested whether central OTR blockade during late gestation disrupts the gestation-related plasticity in intrinsic membrane properties. Whole cell current-clamp recordings were performed in OT neurons from pregnant rats (19–22 days in gestation) that were infused with an OTR antagonist (OTA) or artificial cerebrospinal fluid (aCSF) and from virgin rats infused with aCSF into the third ventricle via an osmotic minipump beginning on days 12–14 of gestation. The amplitudes of both Ca2+-dependent afterhyperpolarizations (AHPs), an apamin-sensitive medium AHP (mAHP) and an apamin-insensitive slow AHP (sAHP), were significantly increased during late gestation in control pregnant animals. However, the amplitude of the sAHP from pregnant rats treated with the OTA was significantly smaller than that of pregnant control rats and similar to that of virgins. These results indicate that the diminished efficiency in lactation due to OTR blockade may be partly a result of an altered sAHP that would shape OT bursting. These findings suggest that central actions of OT during late gestation are necessary for programming the plasticity of at least some of the intrinsic membrane properties in OT neurons during lactation