@prefix dcat: . @prefix dct: . @prefix foaf: . @prefix vcard: . @prefix xsd: . a dcat:Dataset ; dct:description """ The Weinstein equation with complex coefficients is the equation governing axisymmetric potentials (PSA) which can be written as $L_m[u]=\\Delta u+\\left(m/x\\right)\\d_x u =0$, where $m\\in\\C$. This equation is used in particular for modeling the plasma shape in a Tokamak (toroidal chamber with axial magnetic field) for $m = -1$, or it is, when $m=1$, the well-known linearized Ernst equation (which is used to give explicit solutions of the Einstein equations). Here, we generalize results known for $m\\in\\R$ to $m\\in\\C$. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for PSA in the right half-plane $\\H^+$ for Re $m<1$. We establish a new decomposition theorem for the PSA in any annular domains for $m\\in\\C$. In particular, using bipolar coordinates, we prove for annuli (always for $m\\in\\C$) that a family of solutions for PSA equation in terms of associated Legendre functions of first and second kind is complete (the method rests on quasi-separability of variables and some Fourier analysis). For $m\\in\\R$, we show that this family is even a Riesz basis in some non-concentric circular annulus. In the second part, basing on a method due to A. S. Fokas, we give, in explicit integral form, formulas for PSA in a circular domain of the right-half plane $\\H^+$ when $m$ is an integer. These representations are obtained by solving a Riemann-Hilbert problem on the complex plane or on a Riemann surface with two sheets according to the parity of $m$. These formulas involve in an explicit form the Dirichlet and the Neumann data of the PSA in question. In the last part, we study a class of functions which includes the PSA, namely the pseudo-holomorphic functions, i.e. solutions of the complex equation $\\overline{\\partial} w=\\alpha \\overline{w}$, with $\\alpha\\in L^r$, $2\\leq r<\\infty$. We extend the Bers similarity principle (decomposition of pseudo-holomorphic functions in the form $e^s \\,F$ under some regularity assumptions with holomorphic $F$) and a converse of this principle to the critical regularity case $r=2$. Using the connection between pseudo-holomorphic functions and solutions to the conjugate Beltrami equations, we deduce well-posedness of Dirichlet problem in smooth domains with weighted $L^p$ boundary data for 2-D isotropic conductivity equations whose coefficients have logarithm in the Sobolev space $W^{1,2}$. """ ; dct:identifier "tel-00916049" ; dct:issued "2026-05-07T21:32:29.578546"^^xsd:dateTime ; dct:language "fr" ; dct:modified "2026-05-07T21:32:29.578550"^^xsd:dateTime ; dct:publisher ; dct:title "Complex analysis and some Dirichlet problems in the plane: Weinstein's equation and conductivity equation with unbounded coefficients" ; dcat:contactPoint [ a vcard:Organization ; vcard:fn "CCSD" ] ; dcat:distribution ; dcat:keyword "analyse-complexe", "complex-analysis", "equation-de-weinstein", "fonctions-pseudo-holomorphes", "infoeu-reposemanticsdoctoralthesis", "mathmath-apmathematics-mathanalysis-of-pdes-mathap", "mathmath-cvmathematics-mathcomplex-variables-mathcv", "problemes-de-riemann-hilbert", "pseudo-holomorphic-functions", "riemann-hilbert-problems", "sobolev-exposant-critique", "sobolev-space-critical-exponent", "theses", "weinstein-equation" ; dcat:landingPage . a dcat:Distribution ; dct:format "HTML" ; dct:issued "2026-05-07T21:32:29.580565"^^xsd:dateTime ; dct:modified "2026-05-07T21:32:29.560250"^^xsd:dateTime ; dct:title "Complex analysis and some Dirichlet problems in the plane: Weinstein's equation and conductivity equation with unbounded coefficients" ; dcat:accessURL . a foaf:Agent ; foaf:name "test_moissonnage_selune" . a foaf:Document .