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mcd
mcd
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As an addition to Hadrian's answer, for boundary value problems there are as you mention there aresome issues with the boundary and choosing the correct boundary conditions. As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at the boundary. This is also called the Shapiro-Lopatinskii condition. Roughly speaking you define aan operator-valued symbol at the boundary and require that it is invertible.

Reference include: Hörmander 3 (Chapter XX.1) this is somehow the standard viewpoint, "A short introduction to Boutet de Monvel's calculus" by Elmar Schrohe (if you want to consider pseudodifferential boundary value problems), and the book by Egorov and Schulze (they have a readable introduction into elliptic bvp, but most of the book is really tough to read).

As an addition to Hadrian's answer, for boundary value problems as you mention there are issues with the boundary and choosing the correct boundary conditions. As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at the boundary. This is also called the Shapiro-Lopatinskii condition. Roughly speaking you define a operator-valued symbol at the boundary and require that it is invertible.

Reference include: Hörmander 3 (Chapter XX.1) this is somehow the standard viewpoint, "A short introduction to Boutet de Monvel's calculus" by Elmar Schrohe (if you want to consider pseudodifferential boundary value problems), and the book by Egorov and Schulze (they have a readable introduction into elliptic bvp, but most of the book is really tough to read).

As an addition to Hadrian's answer, for boundary value problems there are as you mention some issues with the boundary and choosing the correct boundary conditions. As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at the boundary. This is also called the Shapiro-Lopatinskii condition. Roughly speaking you define an operator-valued symbol at the boundary and require that it is invertible.

Reference include: Hörmander 3 (Chapter XX.1) this is somehow the standard viewpoint, "A short introduction to Boutet de Monvel's calculus" by Elmar Schrohe (if you want to consider pseudodifferential boundary value problems), and the book by Egorov and Schulze (they have a readable introduction into elliptic bvp, but most of the book is really tough to read).

Source Link
mcd
mcd
  • 591
  • 4
  • 12

As an addition to Hadrian's answer, for boundary value problems as you mention there are issues with the boundary and choosing the correct boundary conditions. As a rule of thumb, people want to have a theorem of the kind "ellipticity implies fredholmness". Thus, for bvps you have to add a notion of ellipticity at the boundary. This is also called the Shapiro-Lopatinskii condition. Roughly speaking you define a operator-valued symbol at the boundary and require that it is invertible.

Reference include: Hörmander 3 (Chapter XX.1) this is somehow the standard viewpoint, "A short introduction to Boutet de Monvel's calculus" by Elmar Schrohe (if you want to consider pseudodifferential boundary value problems), and the book by Egorov and Schulze (they have a readable introduction into elliptic bvp, but most of the book is really tough to read).