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Results tagged with elliptic-pde
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user 113756
Questions about partial differential equations of elliptic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
5
votes
Reinforced Maximum Principle
As soon as $n=2$, and $A$ is a symmetric matrix, the answer is surely yes: $u$ has no critical points in $U$. Indeed, we can have a look at what is possibly (to my knowledge) the most recent paper on …
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3
votes
Accepted
Reference request: Parabolic Equations
Premise
This is a long, possibly tedious, comment. In my opinion, the problem with new books on such topics is that in most cases their contents are either expositions of the classical theory or speci …
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5
votes
Accepted
How to prove the second Korn inequality?
You can find a full proof (to my knowledge the simpler one currently known) in the paper [1] and in the book [2], chapter I, §2.1 pp. 14-21. The original proof of Arthur Korn is so long and involved t …
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3
votes
Accepted
Reference request on Pucci extremal operators
The original references are the works of Pucci [2] and [3] which, however, are written in Italian. A perhaps more accessible introduction to these kind of operators is found in the monograph [1], §2.2 …
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2
votes
Contours for harmonic functions in bounded domains
I am not sure if I have correctly understood your question, but it seems to me that you are searching for a method of finding harmonic functions based on tomography/integral geometry: roughly speaking …
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1
vote
Euler-Lagrange equation for a functional
It means that the functional derivative of $J(u)$ is zero (i.e. $J(u)$ has $u$ as a stationary point) if the function $u$ solves the given divergence form PDE, i.e.
$$\DeclareMathOperator{\divg}{\math …
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7
votes
Representing a nonlinear elliptic PDE as an energy minimization problem
As already said by Kosh, you are trying to solve an inverse problem in the calculus of variation: in its classical formulation, given a system of PDE, the problem consists in finding a functional whos …
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19
votes
Accepted
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
This question requires an articulated answer, since the topic dealt is complex and ramified. A fundamental solution for a not necessarily divergence form $2$nd order elliptic system with $C^{2,h}$ coe …
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5
votes
Accepted
Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
Perhaps the best self contained reference on this result is the book [2] by Stampacchia himself. It is a set of typewritten course notes in French, taken from a graduate course on elliptic equations h …
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2
votes
About the Hausdorff dimension of removable singularities of PDE
I am not familiar with the current studies on the Hausdorff dimension of the singular set of solutions to PDE, but I know that in [1] a necessary and sufficient condition for the holding of Hartogs ph …
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