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Pseudo-holomorphic functions at the critical exponent
TLDR
In this article, an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle were proved for the Dirichlet problem with weighted boundary data for 2-D isotropic conductivity equations.Abstract:
We study Hardy classes on the disk associated to the equation $\bar\d w=\alpha\bar w$ for $\alpha\in L^r$ with $2\leq r<\infty$. The paper seems to be the first to deal with the case $r=2$. We prove an analog of the M.~Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in $W^{1,2}$. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for $W^{1,2}_0$-functions.read more
Citations
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Uniqueness results for inverse Robin problems with bounded coefficient
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Massive Scaling Limit of the Ising Model: Subcritical Analysis and Isomonodromy
TL;DR: In this paper, the authors studied the spin n-point functions of the planar Ising model on a simply connected domain \Omega discretised by the square lattice \delta\mathbb{Z}^{2} under nearcritical scaling limit.
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Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane
Slah Chaabi,Stéphane Rigat +1 more
TL;DR: In this article, the authors considered the generalized axisymmetric potentials (GASP) and proved a new decomposition theorem for the GASP in annular domains.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
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Functional Analysis, Sobolev Spaces and Partial Differential Equations
TL;DR: In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.
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