Partial Differential Equations Arising from Physics and Geometry, 2019
Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Reversib... more Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Reversibility in dynamics and group theory, A.G. O'FARRELL & I. SHORT Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢȂ & M. POPA (eds) The Bloch-Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM Arithmetic and geometry, L. DIEULEFAIT et al (eds) O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) Groups St Andrews 2013, C.M. CAMPBELL et al (eds) Inequalities for graph eigenvalues, Z. STANIĆ Surveys in combinatorics 2015, A. CZUMAJ et al (eds) Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) Geometry in a Fréchet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU Sheaves and functions modulo p, L. TAELMAN Recent progress in the theory of the Euler and Navier-Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA Graded rings and graded Grothendieck groups, R. HAZRAT Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds) Random walks and heat kernels on graphs, M.T. BARLOW Evolution equations, K. AMMARI & S. GERBI (eds) Surveys in combinatorics 2017, A. CLAESSON et al (eds) Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD Polynomials and the mod 2 Steenrod algebra II, G. WALKER & R.M.W. WOOD Asymptotic analysis in general relativity, T. DAUDÉ, D. HÄFNER & J.-P. NICOLAS (eds) Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTÍNEZ-PÉREZ & B.E.A. NUCINKIS (eds) Introduction to hidden semi-Markov models, J. VAN DER HOEK & R.J. ELLIOTT Advances in two-dimensional homotopy and combinatorial group theory, W. METZLER & S. ROSEBROCK (eds) New directions in locally compact groups, P.-E. CAPRACE & N. MONOD (eds) Synthetic differential topology, M.C. BUNGE, F. GAGO & A.M. SAN LUIS Permutation groups and cartesian decompositions, C.E. PRAEGER & C. SCHNEIDER Partial differential equations arising from physics and geometry, M. BEN AYED et al (eds) Topological methods in group theory, N. BROADDUS, M. DAVIS, J.-F. LAFONT & I. ORTIZ (eds) Partial differential equations in fluid mechanics, C.L. FEFFERMAN, J.C. ROBINSON & J.L. ROORIGO (eds) Stochastic stability of differential equations in abstract spaces, K.
In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given ... more In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝto have a positive weak solution in Ω with 0 boundary data, which is singular at each x
Les travaux presentes dans cette these portent sur la construction de solutions ayant un lieu sin... more Les travaux presentes dans cette these portent sur la construction de solutions ayant un lieu singulier prescrit pour certaines equations aux derivees partielles elliptiques semi-lineaires. La methode qu'on utilise consiste a definir une famille de solutions approchees au probleme a partir de solutions particulieres radiales, puis a etudier le linearise de l'operateur considere en ces solutions approchees dans des espaces fonctionnels bien choisis en l'occurence les espaces de holder a poids. Enfin, la conclusion est obtenue en utilisant le theoreme des fonctions implicites ou le theoreme du point fixe. Dans le premier article, on construit une solution du probleme avec non-linearite sous-critique de lieu singulier egal a une sous-variete compacte sans bord. Dans le deuxieme article, on s'interesse au cas sur-critique et on montre l'existence de solution faible positive du probleme considere dans la boule unite, ayant une singularite non eliminable en un point fixe proche de l'origine. On donne en particulier une demonstration a un resultat concernant l'equation d'emden enonce par h. Matano. Dans le troisieme article, on generalise le resultat precedent au cas d'un nombre fini de singularites isolees, plus precisement, on montre l'existence d'un ouvert regulier connexe contenant un nombre fini de points fixes et l'existence d'une solution faible positive du probleme qui est singuliere en chacun de ces points. Le quatrieme article de la these est consacre a l'etude du probleme de yamabe singulier. On y montre un resultat de non existence de solution du probleme defini sur un ouvert etoile par rapport a un point qui a une singularite non emilinable en ce point. On y etend aussi les resultats de r. Mazzeo et f. Pacard au cas du probleme de yamabe defini sur un ouvert borne contenant deux points fixes. On donne une condition suffisante portant sur ces deux points pour qu'il existe une solution faible positive du probleme qui est singuliere en chacun de ces points.
Given 0 any open and bounded subset of R n , n 4, with smooth boundary and given 7 any (n&m)-dime... more Given 0 any open and bounded subset of R n , n 4, with smooth boundary and given 7 any (n&m)-dimensional compact submanifold of 0 without boundary, n>m>2, we prove the existence of weak solutions to the problem &2u=u p in 0 { u>0 in 0 u=0 on 0, which are singular on 7, when p is a real p>mÂ(m&2), close to this value. 1996 Academic Press, Inc. &2u=u (n+2)Â(n&2) .
If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq... more If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq 3$ or if $f$ is given by $e^u$ when $N=3$, we prove the existence of a positive weak solution of $ \Delta u + \lambda f(u) =0 $ which is defined in the unit ball of ${\Bbb R}^N$, has $0$ boundary data and has a nonremovable prescribed singularity at some point $x_0$ close to the origin.
In this paper, we are interested in the following biharmonic equation:with Navier boundary condit... more In this paper, we are interested in the following biharmonic equation:with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝ
We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlin... more We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up pointâ, we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of "profile order", we show that it is upper semicontinuous, and continuous only at points where it is a local minimum. Résumé Nous considérons des solutions explosives de l'équation semilinéaire de la chaleur avec une nonlinéarité sous-critique au sens de Sobolev. Etant donné un point d'explosionâ, grâceà des travaux antérieurs, on connaît le comportement asymptotique des solutions en variables auto-similaires. Notre objectif est de discuter la stabilité de ce comportement, par rapportà des perturbations du point d'explosion et de la donnée initiale. Introduisant la notion de "l'ordre du profil", nous montrons qu'il est semicontinu supérieurement, et continu uniquement aux points où il est un minimum local.
We consider the fundamental solution GaGa of the operator −Δa=−1a(x)div(a(x)∇⋅) on a bounded smoo... more We consider the fundamental solution GaGa of the operator −Δa=−1a(x)div(a(x)∇⋅) on a bounded smooth domain Ω⊂RnΩ⊂Rn (n⩾2n⩾2), associated to the Dirichlet boundary condition, where a is a positive smooth function on Ω¯. In this short Note, we give a precise description of the function Ga(x,y)Ga(x,y). In particular, we define in a unique way its continuous part Ha(x,y)Ha(x,y) and we prove that the corresponding Robin's function Ra(x)=Ha(x,x)Ra(x)=Ha(x,x) belongs to C∞(Ω)C∞(Ω), although Ha∉C1(Ω×Ω)Ha∉C1(Ω×Ω) in general.
Partial Differential Equations Arising from Physics and Geometry, 2019
Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Reversib... more Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) Reversibility in dynamics and group theory, A.G. O'FARRELL & I. SHORT Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢȂ & M. POPA (eds) The Bloch-Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM Arithmetic and geometry, L. DIEULEFAIT et al (eds) O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) Groups St Andrews 2013, C.M. CAMPBELL et al (eds) Inequalities for graph eigenvalues, Z. STANIĆ Surveys in combinatorics 2015, A. CZUMAJ et al (eds) Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) Geometry in a Fréchet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU Sheaves and functions modulo p, L. TAELMAN Recent progress in the theory of the Euler and Navier-Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA Graded rings and graded Grothendieck groups, R. HAZRAT Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds) Random walks and heat kernels on graphs, M.T. BARLOW Evolution equations, K. AMMARI & S. GERBI (eds) Surveys in combinatorics 2017, A. CLAESSON et al (eds) Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD Polynomials and the mod 2 Steenrod algebra II, G. WALKER & R.M.W. WOOD Asymptotic analysis in general relativity, T. DAUDÉ, D. HÄFNER & J.-P. NICOLAS (eds) Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTÍNEZ-PÉREZ & B.E.A. NUCINKIS (eds) Introduction to hidden semi-Markov models, J. VAN DER HOEK & R.J. ELLIOTT Advances in two-dimensional homotopy and combinatorial group theory, W. METZLER & S. ROSEBROCK (eds) New directions in locally compact groups, P.-E. CAPRACE & N. MONOD (eds) Synthetic differential topology, M.C. BUNGE, F. GAGO & A.M. SAN LUIS Permutation groups and cartesian decompositions, C.E. PRAEGER & C. SCHNEIDER Partial differential equations arising from physics and geometry, M. BEN AYED et al (eds) Topological methods in group theory, N. BROADDUS, M. DAVIS, J.-F. LAFONT & I. ORTIZ (eds) Partial differential equations in fluid mechanics, C.L. FEFFERMAN, J.C. ROBINSON & J.L. ROORIGO (eds) Stochastic stability of differential equations in abstract spaces, K.
In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given ... more In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝto have a positive weak solution in Ω with 0 boundary data, which is singular at each x
Les travaux presentes dans cette these portent sur la construction de solutions ayant un lieu sin... more Les travaux presentes dans cette these portent sur la construction de solutions ayant un lieu singulier prescrit pour certaines equations aux derivees partielles elliptiques semi-lineaires. La methode qu'on utilise consiste a definir une famille de solutions approchees au probleme a partir de solutions particulieres radiales, puis a etudier le linearise de l'operateur considere en ces solutions approchees dans des espaces fonctionnels bien choisis en l'occurence les espaces de holder a poids. Enfin, la conclusion est obtenue en utilisant le theoreme des fonctions implicites ou le theoreme du point fixe. Dans le premier article, on construit une solution du probleme avec non-linearite sous-critique de lieu singulier egal a une sous-variete compacte sans bord. Dans le deuxieme article, on s'interesse au cas sur-critique et on montre l'existence de solution faible positive du probleme considere dans la boule unite, ayant une singularite non eliminable en un point fixe proche de l'origine. On donne en particulier une demonstration a un resultat concernant l'equation d'emden enonce par h. Matano. Dans le troisieme article, on generalise le resultat precedent au cas d'un nombre fini de singularites isolees, plus precisement, on montre l'existence d'un ouvert regulier connexe contenant un nombre fini de points fixes et l'existence d'une solution faible positive du probleme qui est singuliere en chacun de ces points. Le quatrieme article de la these est consacre a l'etude du probleme de yamabe singulier. On y montre un resultat de non existence de solution du probleme defini sur un ouvert etoile par rapport a un point qui a une singularite non emilinable en ce point. On y etend aussi les resultats de r. Mazzeo et f. Pacard au cas du probleme de yamabe defini sur un ouvert borne contenant deux points fixes. On donne une condition suffisante portant sur ces deux points pour qu'il existe une solution faible positive du probleme qui est singuliere en chacun de ces points.
Given 0 any open and bounded subset of R n , n 4, with smooth boundary and given 7 any (n&m)-dime... more Given 0 any open and bounded subset of R n , n 4, with smooth boundary and given 7 any (n&m)-dimensional compact submanifold of 0 without boundary, n>m>2, we prove the existence of weak solutions to the problem &2u=u p in 0 { u>0 in 0 u=0 on 0, which are singular on 7, when p is a real p>mÂ(m&2), close to this value. 1996 Academic Press, Inc. &2u=u (n+2)Â(n&2) .
If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq... more If $f$ is either given by $(1+u)^p$ for some $\frac{N+2}{N-2}< p < \frac{N+1}{N-3}$, $N\geq 3$ or if $f$ is given by $e^u$ when $N=3$, we prove the existence of a positive weak solution of $ \Delta u + \lambda f(u) =0 $ which is defined in the unit ball of ${\Bbb R}^N$, has $0$ boundary data and has a nonremovable prescribed singularity at some point $x_0$ close to the origin.
In this paper, we are interested in the following biharmonic equation:with Navier boundary condit... more In this paper, we are interested in the following biharmonic equation:with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝ
We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlin... more We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up pointâ, we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of "profile order", we show that it is upper semicontinuous, and continuous only at points where it is a local minimum. Résumé Nous considérons des solutions explosives de l'équation semilinéaire de la chaleur avec une nonlinéarité sous-critique au sens de Sobolev. Etant donné un point d'explosionâ, grâceà des travaux antérieurs, on connaît le comportement asymptotique des solutions en variables auto-similaires. Notre objectif est de discuter la stabilité de ce comportement, par rapportà des perturbations du point d'explosion et de la donnée initiale. Introduisant la notion de "l'ordre du profil", nous montrons qu'il est semicontinu supérieurement, et continu uniquement aux points où il est un minimum local.
We consider the fundamental solution GaGa of the operator −Δa=−1a(x)div(a(x)∇⋅) on a bounded smoo... more We consider the fundamental solution GaGa of the operator −Δa=−1a(x)div(a(x)∇⋅) on a bounded smooth domain Ω⊂RnΩ⊂Rn (n⩾2n⩾2), associated to the Dirichlet boundary condition, where a is a positive smooth function on Ω¯. In this short Note, we give a precise description of the function Ga(x,y)Ga(x,y). In particular, we define in a unique way its continuous part Ha(x,y)Ha(x,y) and we prove that the corresponding Robin's function Ra(x)=Ha(x,x)Ra(x)=Ha(x,x) belongs to C∞(Ω)C∞(Ω), although Ha∉C1(Ω×Ω)Ha∉C1(Ω×Ω) in general.
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