Distributing many points on a sphere

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

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The Porous Medium Equation: Mathematical Theory

The porous medium equation

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and

The Porous Medium Equation

Estimates of harmonic measure

On donne des resultats optimaux pour la resolubilite du probleme de Neumann dans des domaines de Lipschitz a donnees dans L p . On obtient des resultats du point extreme correspondant pour des espaces de Hardy

Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains

On the absolute continuity of elliptic measures

On etudie le probleme de Dirichlet pour l'operateur biharmonique Δ 2 , dans un domaine borne lipschitzien quelconque D dans R n , et on donne des bornes optimales. On demontre des resultats d'existence et d'unicite quand les valeurs au bord ont des derivees dans L 2 ∂D), et la derivee normale appartient a L 2 (∂D). La solution obtenue prend les valeurs au bord dans le sens de la convergence non tangentielle, et la fonction maximale non tangentielle du ⊇u appartient a L 2 (∂D)

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The Dirichlet problem for the biharmonic equation in a Lipschitz domain

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Björn E. J. Dahlberg

Papers

Estimates of harmonic measure

Hardy spaces and the Neumann problem in L^p for laplace's equation in Lipschitz domains

On the absolute continuity of elliptic measures

The Dirichlet problem for the biharmonic equation in a Lipschitz domain

Area integral estimates for higher order elliptic equations and systems