Luis Silvestre
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Registered User
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Jun 17 |
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Does this sequence of H\"older functions have a limit? Why wouldn't you remove the "$4,$" in all the exponents? |
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Jun 6 |
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Reference Search for a Functional Minimization Problem Are you sure you have $u^3$ and not $u^4$ in the functional? I'm saying because Cahn-Hilliard is the gradient flow of the functional with $u^4$. The minimizers of that functional (with $u^4$) are well understood (Google for "De Giorgi's conjecture") and are definitely not compactly supported. If you stick to $u^3$, the equation you have is the one that you would get for traveling waves in the KPP-Fisher equation. Those are well understood as well. They are certainly not compactly supported either. |
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Jun 6 |
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Regularity of the right hand side (the source term) in Evans-Krylov theory Check Theorem 6.6 in the book of Caffarelli and Cabre for the version with zero right hand side. The case of $g \in C^\alpha$ follows from Theorem 8.1 in the same book after you decipher those hypothesis. |
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Jun 5 |
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Regularity of the right hand side (the source term) in Evans-Krylov theory This is correct. Also, it is not necessary to assume $u$ to have Lipschitz gradient or $F \in C^2$. I think that the Holder condition on the growth of the $L^n$ norm of $g$ that you suggest is equivalent to $g \in C^\alpha$ by Campanato's theorem. |
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Apr 30 |
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maximum principle for a non-uniformly parabolic operator What do you mean by the maximum principle? If you want the maximum of the solution to be decreasing in time, then it would not be true (even replacing the factor $e^{-\beta t}$ by $1$). If you want a comparison principle saying that if one solution is initially larger than another, then the order is preserved by time, then that will be true. If you want a bound on the $L^\infty$ norm for a solution $u$, then $||u(\cdot,t)||_{L^\infty} \leq e^{-(\min G_x)t} ||u(\cdot,0)||_{L^\infty}$ even if the second order term wasn't there. |
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Apr 9 |
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Rewriting the advection-diffusion equation The first particular case to think of is when u is equal to zero and K is the identity. In that case p solves the usual heat equation. Do we get any useful insight for the heat equation from this dynamical viewpoint? |
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Mar 8 |
accepted | Regularity of parabolic equation; divergence free drift |
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Mar 7 |
answered | Regularity of parabolic equation; divergence free drift |
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Feb 8 |
accepted | Nash’s paper on parabolic equations. |
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Feb 7 |
answered | Nash’s paper on parabolic equations. |
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Feb 5 |
accepted | Boundedness of a given boundary value problem. |
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Feb 5 |
answered | Boundedness of a given boundary value problem. |
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Jan 16 |
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Viscosity solution of the PDE Yes, but the proof in math.stackexchange is wrong because the functions $\phi$ may not exist if $u_1$ and $u_2$ are not differentiable at $x_0$. The classical proof uses the doubling of variables method, very much like in the proof of Theorem 1 in section 10.2 of Evans book. |
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Jan 15 |
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Viscosity solution of the PDE This is literally the first example I saw when I studied viscosity solutions in grad school. Have you checked the users guide to viscosity solutions for a general existence and unique theorem? |
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Jan 8 |
awarded | ● Necromancer |
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Jan 8 |
awarded | ● Commentator |
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Jan 8 |
revised |
Counterexamples in PDE deleted 18 characters in body |
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Jan 8 |
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Counterexamples in PDE This (amazing) result is more a statement about the definition of weak solutions than about the Euler equation as a model. In fact, in 2D, the Euler equation is well posed in the classical sense. |
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Jan 8 |
answered | Counterexamples in PDE |
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Jan 2 |
accepted | variation of the obstacle in the obstacle problem |
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Dec 24 |
answered | Are $\lVert \Delta u \rVert_{L^2(S)}$ and $\lVert u \rVert_{H^2(S)}$ equivalent norms on a compact manifold? |
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Dec 24 |
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variation of the obstacle in the obstacle problem It may also be worth mentioning that the Wikipedia page on the obstacle problem is not bad. |
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Dec 24 |
answered | variation of the obstacle in the obstacle problem |
