INMABB   05456
INSTITUTO DE MATEMATICA BAHIA BLANCA
Unidad Ejecutora - UE
libros
Título:
REMARKS ON A THEOREM OF A. PLEIJEL and related topics, II
Autor/es:
AGNES BENEDEK; RAFAEL PANZONE
Editorial:
Universidad Nacional del Sur
Referencias:
Lugar: Bahía Blanca; Año: 2007 p. 101
Resumen:
REMARKS ON A THEOREM OF Å. PLEIJEL AND RELATED TOPICS, II, ON THE NEUMANN BOUNDARY PROBLEM FOR A PLANE JORDAN REGION. Agnes Benedek and Rafael Panzone Instituto de Matemática (INMABB, UNS-CONICET), Alem 1253, (8000) Bahía Blanca, ARGENTINA. ON THE NEUMANN BOUNDARY PROBLEM FOR A PLANE JORDAN REGION. Agnes Benedek and Rafael Panzone Instituto de Matemática (INMABB, UNS-CONICET), Alem 1253, (8000) Bahía Blanca, ARGENTINA. ON THE NEUMANN BOUNDARY PROBLEM FOR A PLANE JORDAN REGION. Agnes Benedek and Rafael Panzone Instituto de Matemática (INMABB, UNS-CONICET), Alem 1253, (8000) Bahía Blanca, ARGENTINA. Å. PLEIJEL AND RELATED TOPICS, II, ON THE NEUMANN BOUNDARY PROBLEM FOR A PLANE JORDAN REGION. Agnes Benedek and Rafael Panzone Instituto de Matemática (INMABB, UNS-CONICET), Alem 1253, (8000) Bahía Blanca, ARGENTINA. RESUMEN. El problema de Neumann en una región de Jordan (acotada) plana D de contorno J suficientemente regular admite una sucesión de autovalores y autofunciones, , en J, en D, , tales que si J es vale en (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente, (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente, (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente,en J, en D, , tales que si J es vale en , en J, en D, , tales que si J es vale en (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente,en J, en D, , tales que si J es vale en (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente, (**) . Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. . Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. ABSTRACT. We consider Neumann´s problem for the Laplacian in a plane Jordan (bounded) region D with regular boundary J. If is the set of eigenvalues of that problem, the counting function satisfies H. Weyl asymptotic formula: . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . We consider Neumann´s problem for the Laplacian in a plane Jordan (bounded) region D with regular boundary J. If is the set of eigenvalues of that problem, the counting function satisfies H. Weyl asymptotic formula: . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . =. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . In this paper we collect some results on eigenvalues, eigenfunctions and Green´s kernel that hold for plane regular membranes. We show that the preceding equality holds for a Jordan region with a -boundary and present a simplified proof for this case without using Weyl´s asymptotic formula. In fact, it is a consequence of Pleijel´s formula. Besides we show that the variational eigenvalues and eigenfunctions and the classical ones coincide for general Jordan regions proving then that in our framework we can use the results obtained by the powerful variational method. This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 We show that the preceding equality holds for a Jordan region with a -boundary and present a simplified proof for this case without using Weyl´s asymptotic formula. In fact, it is a consequence of Pleijel´s formula. Besides we show that the variational eigenvalues and eigenfunctions and the classical ones coincide for general Jordan regions proving then that in our framework we can use the results obtained by the powerful variational method. This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 El problema de Neumann en una región de Jordan (acotada) plana D de contorno J suficientemente regular admite una sucesión de autovalores y autofunciones, , en J, en D, , tales que si J es vale en (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente,en J, en D, , tales que si J es vale en (*) , g(z) holomorfa en el semiplano derecho. Pleijel demuestra esta fórmula para curvas . Sea . (*) puede demostrarse sin utilizar la siguiente fórmula asintótica de H. Weyl que es consecuencia de la precedente, (**) . Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. . Los planteos clásico y variacional dan lugar a los mismos autovalores con las mismas autofunciones. Estas son también autofunciones del operador de Green y tienen otras propiedades además de las indicadas. La exposición es independiente del vol. I y autocontenida. Palabras clave: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral.: problema de Neumann, autofunciones, autovalores, serie de Dirichlet espectral. ABSTRACT. We consider Neumann´s problem for the Laplacian in a plane Jordan (bounded) region D with regular boundary J. If is the set of eigenvalues of that problem, the counting function satisfies H. Weyl asymptotic formula: . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . We consider Neumann´s problem for the Laplacian in a plane Jordan (bounded) region D with regular boundary J. If is the set of eigenvalues of that problem, the counting function satisfies H. Weyl asymptotic formula: . Related to the monotonous function is the spectral Dirichlet series: P(z)==. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . , g(z) holomorphic at least in . , g(z) holomorphic at least in . =. The behaviour of P(z) follows from that of but results about P(z) can be obtained without a priori knowledge of the counting function. A theorem of Å. Pleijel deals with this type of results. He proves that if D has a -boundary the following formula holds: , g(z) holomorphic at least in . In this paper we collect some results on eigenvalues, eigenfunctions and Green´s kernel that hold for plane regular membranes. We show that the preceding equality holds for a Jordan region with a -boundary and present a simplified proof for this case without using Weyl´s asymptotic formula. In fact, it is a consequence of Pleijel´s formula. Besides we show that the variational eigenvalues and eigenfunctions and the classical ones coincide for general Jordan regions proving then that in our framework we can use the results obtained by the powerful variational method. This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for solutions of elliptic partial differential equations. E. Hopf´s lemma. Fundamental solution for the metaharmonic operator . -harmonic functions (metaharmonic functions). Phrágmen-Lindelöff maximum principle. Removable singularities. Poisson´s kernel. Maximum principle for -harmonic functions. Mean-value theorem. A. Harnack´s inequality. Uniqueness of the solution of Neumann´s problem for the metaharmonic differential operator. 23 CHAPTER 6. Neumann problem for the metaharmonic differential operator. Simple layer potential. Double layer potential.. Solution of the problem: existence and uniqueness. Properties of the solution. 34 CHAPTER 7. Green´s kernel for the metaharmonic operator . Properties of the Green´s kernel. Auxiliary results: data lemma, area lemma, basic lemma and boundary lemma. Green´s operator: . Its relation with the equation . 47 CHAPTER 8. Eigenfunctions of Green´s operator. Admissible functions. Equivalence of the systems of classical eigenfunctions and variational ones for Jordan regions. Properties of Green´s operator. 61 CHAPTER 9. Theorems of Å. Pleijel, H. Weyl and S. Ikehara. 69 CHAPTER 10. NOTES 82 BIBLIOGRAPHY 97 INDEX 99 ERRATA 101 This work is essentially of expository nature and mainly self-contained. No use is made of volume I. Its central core is in Chapters 5-9 that can be read almost independently of the first four chapters. Key words: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series.: Neumann problem, eigenvalue, eigenfunction, spectral Dirichlet series. CONTENTS CHAPTER 1. Introduction. Classical Neumann eigenvalue problem for plane (bounded) Jordan regions. H. Weyl´s theorem. Quasidisks. H. Whitney´s theorem. P Jones´ extension operators. Variational Neumann eigenvalue problem. Weak solutions. Eigenvalues and eigenfunctions. Fredholm alternative. 1 CHAPTER 2. Eigenvalues: characterization and infsup (minmax) properties. Eigenfunctions: completeness theorem. Boundary of a quasidisk. Distribution of the eigenvalues. 7 CHAPTER 3. Triplets . The counting functions and . 13 CHAPTER 4. Variational triplets and the Neumann problem. The distribution function (counting function) and its relation with and . 18 CHAPTER 5. Normal derivatives. Maximum principle for