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Results tagged with ap.analysis-of-pdes
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user 90154
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
11
votes
0
answers
339
views
Elliptic regularity of perturbed scalar curvature in Kazdan & Warner
In their paper A Direct Approach to the Determination of Gaussian
and Scalar Curvature Functions, Kazdan and Warner claim something along the lines of: if $g$ is a metric in $W^{2,p}$ ($p>n$) whose sc …
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10
votes
1
answer
419
views
Does harmonic map heat flow of a curve always fully converge to a geodesic?
Consider a smooth closed curve $u_0$ in a compact Riemannian manifold $(M,g)$. Let $u_0$ evolve by harmonic map heat flow, $\partial_tu=\nabla_{\partial_su}\partial_su$, and call the result $u(t)$.
…
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4
votes
0
answers
182
views
Traces of manifold-valued Sobolev maps
Let $(M^m,g)$ be a compact Riemannian manifold with smooth nonempty boundary, and $N^n\subseteq \Bbb R^d$ a boundaryless isometrically embedded Riemannian manifold. For $1\le p<\infty$ we define as us …
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4
votes
Dirichlet problem for manifold, how to prove $W^{1,2}_0(\Omega)$ solution is $C^{2,\alpha}(\...
I will essentially explain the comment under Theorem 8.14 in Gilbarg-Trudinger.
I will assume the result stated there: given smooth boundary data and RHS, the Poisson equation has a unique smooth so …
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2
votes
0
answers
155
views
Ricci flow with surgery without the "no locally separating $\Bbb RP^2$" assumption
In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ w …
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1
vote
2
answers
763
views
Proof of the du Bois-Reymond lemma "by approximation" [closed]
I'm curious about the following argument in Morrey ("Multiple integrals in the calculus of variations", Lemma 2.3.1). Suppose $f\in L^1[0,1]$ and $$\int_0^1 fg\,dx=0$$ for every test function $g\in C^ …
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1
vote
0
answers
224
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Weak elliptic maximum principle on manifolds without strict ellipticity
This question is not to be confused with the similarly titled question here.
In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ellipt …
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