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The text is intended for students who wish a concise and rapid introduction to some main topics in PDEs, necessary for understanding current research, especially in nonlinear PDEs. Organized on three parts, the book guides the reader from fundamental classical results, to some aspects of the modern theory and furthermore, to some techniques of nonlinear analysis. Compared to other introductory books in PDEs, this work clearly explains the transition from classical to generalized solutions and the natural way in which Sobolev spaces appear as completions of spaces of continuously differentiable functions with respect to energetic norms. Also, special attention is paid to the investigation of the solution operators associated to elliptic, parabolic and hyperbolic non-homogeneous equations anticipating the operator approach of nonlinear boundary value problems. Thus the reader is made to understand the role of linear theory for the analysis of nonlinear problems.

Differential equations, Linear --- Differential equations, Partial --- Linear differential equations --- Linear systems --- Partial differential equations --- Elliptic Equation. --- Heat Equation. --- Partial Differential Equation. --- Sobolev Space. --- Wave Equation.

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Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

Differential equations, Elliptic. --- Differential equations, Partial. --- Partial differential equations --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- Dirichlet Problems for Elliptic Equations. --- Distributional Solution. --- Entropy Solutions. --- Leray-Lions Problem. --- Nonlinear Elliptic Partial Differential Equations in Divergence Form. --- Sobolev Space.

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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.

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This book includes 13 papers concerning some of the recent progress in the theory of function spaces and its applications. The involved function spaces include Morrey and weak Morrey spaces, Hardy-type spaces, John–Nirenberg spaces, Sobolev spaces, and Besov and Triebel–Lizorkin spaces on different underlying spaces, and they are applied in the study of problems ranging from harmonic analysis to potential analysis and partial differential equations, such as the boundedness of paraproducts and Calderón operators, the characterization of pointwise multipliers, estimates of anisotropic logarithmic potential, as well as certain Dirichlet problems for the Schrödinger equation.

Research & information: general --- Mathematics & science --- expansive matrix --- (mixed-norm) Hardy space --- molecule --- Calderón–Zygmund operator --- real interpolation --- besov space --- meyer wavelet --- Euclidean space --- cube --- congruent cube --- BMO --- JNp --- (localized) John–Nirenberg–Campanato space --- Riesz–Morrey space --- vanishing John–Nirenberg space --- duality --- commutator --- commutators --- Riesz potential --- homogeneous group --- space of homogeneous type --- paraproduct --- T(1) theorem --- hardy space --- bilinear estimate --- Hajłasz–Sobolev space --- Hajłasz–Besov space --- Hajłasz–Triebel–Lizorkin space --- generalized smoothness --- Lebesgue point --- capacity --- pointwise multipliers --- Morrey spaces --- block spaces --- convexification --- Calderón operator --- Hardy’s inequality --- variable Lebesgue space --- local Morrey space --- local block space --- extrapolation --- anisotropy --- Hardy space --- continuous ellipsoid cover --- maximal function --- anisotropic log-potential --- optimal polynomial inequality --- annulus body --- dual log-mixed volume --- Sobolev spaces --- compact manifolds --- tensor bundles --- differential operators --- Triebel–Lizorkin space --- Hardy inequality --- uniform domain --- fractional Laplacian --- Schrödinger equation --- Morrey space --- Dirichlet problem --- metric measure space --- expansive matrix --- (mixed-norm) Hardy space --- molecule --- Calderón–Zygmund operator --- real interpolation --- besov space --- meyer wavelet --- Euclidean space --- cube --- congruent cube --- BMO --- JNp --- (localized) John–Nirenberg–Campanato space --- Riesz–Morrey space --- vanishing John–Nirenberg space --- duality --- commutator --- commutators --- Riesz potential --- homogeneous group --- space of homogeneous type --- paraproduct --- T(1) theorem --- hardy space --- bilinear estimate --- Hajłasz–Sobolev space --- Hajłasz–Besov space --- Hajłasz–Triebel–Lizorkin space --- generalized smoothness --- Lebesgue point --- capacity --- pointwise multipliers --- Morrey spaces --- block spaces --- convexification --- Calderón operator --- Hardy’s inequality --- variable Lebesgue space --- local Morrey space --- local block space --- extrapolation --- anisotropy --- Hardy space --- continuous ellipsoid cover --- maximal function --- anisotropic log-potential --- optimal polynomial inequality --- annulus body --- dual log-mixed volume --- Sobolev spaces --- compact manifolds --- tensor bundles --- differential operators --- Triebel–Lizorkin space --- Hardy inequality --- uniform domain --- fractional Laplacian --- Schrödinger equation --- Morrey space --- Dirichlet problem --- metric measure space

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The ∂̄ Neumann problem is probably the most important and natural example of a non-elliptic boundary value problem, arising as it does from the Cauchy-Riemann equations. It has been known for some time how to prove solvability and regularity by the use of L2 methods. In this monograph the authors apply recent methods involving the Heisenberg group to obtain parametricies and to give sharp estimates in various function spaces, leading to a better understanding of the ∂̄ Neumann problem. The authors have added substantial background material to make the monograph more accessible to students.Originally published in 1977.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Partial differential equations --- Neumann problem. --- Neumann problem --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Boundary value problems --- Differential equations, Partial --- A priori estimate. --- Abuse of notation. --- Analytic continuation. --- Analytic function. --- Approximation. --- Asymptotic expansion. --- Asymptotic formula. --- Basis (linear algebra). --- Besov space. --- Boundary (topology). --- Boundary value problem. --- Boundedness. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Characterization (mathematics). --- Combination. --- Commutative property. --- Commutator. --- Complex analysis. --- Complex manifold. --- Complex number. --- Computation. --- Convolution. --- Coordinate system. --- Corollary. --- Counterexample. --- Derivative. --- Determinant. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Dimensional analysis. --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic boundary value problem. --- Equation. --- Error term. --- Estimation. --- Even and odd functions. --- Existential quantification. --- Function space. --- Fundamental solution. --- Green's theorem. --- Half-space (geometry). --- Hardy's inequality. --- Heisenberg group. --- Holomorphic function. --- Infimum and supremum. --- Integer. --- Integral curve. --- Integral expression. --- Inverse function. --- Invertible matrix. --- Iteration. --- Laplace's equation. --- Left inverse. --- Lie algebra. --- Lie group. --- Linear combination. --- Logarithm. --- Lp space. --- Mathematical induction. --- Neumann boundary condition. --- Notation. --- Open problem. --- Orthogonal complement. --- Orthogonality. --- Parametrix. --- Partial derivative. --- Pointwise. --- Polynomial. --- Principal branch. --- Principal part. --- Projection (linear algebra). --- Pseudo-differential operator. --- Quantity. --- Recursive definition. --- Schwartz space. --- Scientific notation. --- Second derivative. --- Self-adjoint. --- Singular value. --- Sobolev space. --- Special case. --- Standard basis. --- Stein manifold. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Tangent bundle. --- Theorem. --- Theory. --- Upper half-plane. --- Variable (mathematics). --- Vector field. --- Volume element. --- Weak solution. --- Neumann, Problème de --- Equations aux derivees partielles --- Problemes aux limites