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Everything about Elliptic operator
Type of differential operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Elliptic operatorA 5-minute orientation
- Skim a recent paper: Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential EquationsInfluential work since 2022
- Pick up a classic book: Théorie spectrale pour des opérateurs globalement elliptiquesThe most-reprinted book on the subject
- Read the landmark paper: The Index of Elliptic Operators: IIICited 825 times
- Go sideways: Gårding's inequalityA closely related topic
- Go sideways: Schauder estimatesA closely related topic
📖 Definitions
Noun Any of a particular class of second-order linear differential operators.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
The Index of Elliptic Operators: III
The spectral function of an elliptic operator
Adaptive frame methods for elliptic operator equations
Data-sparse approximation to the operator-valued functions of elliptic operator
Self-adjoint quadratic operator pencils and elliptic problems
Equivalent operator preconditioning for elliptic problems
Perturbation algebra of an elliptic operator
A Comparison Theorem for an Elliptic Operator
🆕 Influential since 2022
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
Solving parametric elliptic interface problems via interfaced operator network
Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs
On an Elliptic Operator Degenerating on the Boundary
A general elliptic equation with intrinsic operator
📚 Books worth reading
- 📕Théorie spectrale pour des opérateurs globalement elliptiquesBernard Helffer · 1981
- 📕Lectures on Exponential Decay of Solutions of Second-Order Elliptic EquationsShmuel Agmon · 2014
- 📕Index theory of elliptic boundary problemsStephan Rempel · 1982
- 📕Hörmander spaces, interpolation, and elliptic problemsVladimir A. Mikhailets · 2014
Extremum problems for eigenvalues of elliptic operatorsAntoine Henrot · 2006
NONLINEAR ANALYSIS AND SEMILINEAR ELLIPTIC PROBLEMSANTONIO AMBROSETTI · 2007
Heat kernels and analysis on manifolds, graphs, and metric spacesPascal Auscher · 2003
Elliptic theory on singular manifoldsVladimir E. Nazaikinskii, Anton Yu. Savin · 2005
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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