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Results tagged with ap.analysis-of-pdes
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user 16659
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4
votes
Accepted
Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions
Yes, Safonov showed here that, for any $\alpha \in (0,\,1)$, there is a function $u$ on $\mathbb{R}^3$ that is homogeneous of degree $\alpha$ (in particular has unbounded gradient) and solves a linear …
- 4,899
5
votes
Accepted
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
Yes, this inequality is true, and in fact the assumption about the smallness of $D^2u$ in $L^{2p}$ is not needed.
Using that $|\nabla u|$ is $\epsilon$-close to one in $L^1(B_2)$, we can find a point …
- 4,899
6
votes
Are all positive eigenfunctions principal eigenfunctions?
Another argument perhaps worth mentioning uses the maximum principle. To illustrate the idea we assume that $\Omega$ is a smooth bounded domain, that $L = \Delta$, and that we are dealing with Dirichl …
- 4,899
6
votes
$C^0$ estimate for solutions of elliptic PDE with Neumann BC
Yes, I think the estimate you propose is true. As a simple case take $f = 0$
and $M = B_1 \subset \mathbb{R}^n$. By adding a constant assume that $\inf_{B_1} u = 0$ and $\sup_{B_1}u = K,$ and note tha …
- 144
3
votes
Regularity of Newtonian potential along smooth boundary
One more perspective: after performing a diffeomorphism that locally flattens the boundary, we get an equation of the form
$$a_{ij}(x)w_{ij} + b_i(x)w_i = f(x_n) \text{ in } B_1,$$
where the coefficie …
- 4,899
3
votes
Accepted
'Degenerate' tangent point of a minimal graph
Yes, this is possible. Consider the intersection of the helicoid $z = \tan^{-1}(y/x)$ with its tangent plane $z = y$ at $(1,0,0)$. The projection of the intersection curves to $\{z = 0\}$ consists of …
- 4,899
4
votes
Optimal assumption on H^2 regularity
One cannot weaken the regularity hypothesis on $A$ to $C^{\alpha}$. Consider for example the 1D case: when $f = 0$ the solution $u$ satisfies
$$u'(x) = \frac{const.}{A(x)},$$
which is not in $H^1$ for …
- 4,899
4
votes
Accepted
Doubts in first lemma in the paper of Adams regarding sharp Moser inequality
For the second, rewrite $F(t) \leq \lambda$ as
$$t - \lambda \leq \left(\int_{\mathbb{R}} a(s,\,t)\phi(s)\,ds\right)^q,$$
which reduces the problem to showing that
$$\int_{\mathbb{R}} a(s,\,t)\phi(s)\ …
- 4,899
5
votes
Accepted
Bernstein's corollary for the case of half space
Here is a counterexample: let
$$u(x,y) = e^{-x^2}\sinh(y).$$
Then
$$\det D^2u = -2e^{-2x^2}(\sinh^2(y) + 2x^2) < 0 \text{ ơn } \mathbb{R}^2 \backslash \{0\},$$
and the equation
$$u_{xx} + (2-4x^2)u_{y …
- 4,899
3
votes
Accepted
Estimates on the second-order derivatives for degenerate Monge-Ampere equations
This inequality comes from scaling. Assume that $f$ satisfies $|D^2f| \leq 1$ on $\mathbb{R}^n$ and $f \geq 0$. It suffices to prove that
$$|\nabla f(0)|^2 \leq 2f(0).$$
Equality holds if $f(0) = 0$, …
- 4,899
4
votes
Accepted
Reference request: continuity of the derivatives of the (fundamental) solution to a paraboli...
The equation for $q = p_x$ can be written in divergence form as
$$q_t + b(t)q_x + (D(t,\,x)q_x)_x = 0,$$
so Nash's theorem (which applies to divergence-form equations) implies that $p_x$ is Holder con …
- 4,899
5
votes
Higher regularity of solutions of non-linear elliptic PDE
It is true and well-known (assuming $F$ is e.g. uniformly elliptic). The idea is sketched in ch. 9 of the book by Caffarelli-Cabre, but I am not sure of a precise reference at this level of generality …
- 4,899
8
votes
Points where harmonic functions fail to give a coordinates system
To rule out the trivial case that one of the $f_j$ is a nonzero constant, I interpret the linear independence condition as saying that no nontrivial linear combination of the $f_j$ is constant.
The se …
- 4,899
5
votes
Vanishing rate of a harmonic function near a boundary point
There is a counterexample. Consider the harmonic function
$$u(x,y) = Re\left(e^{-1/z^2}\right) = e^{-\frac{x^2-y^2}{r^4}}\cos\left(\frac{2xy}{r^4}\right),$$
where $r^2 = x^2 + y^2$. We have that
$$u(x …
- 4,899
4
votes
Accepted
Does the difference of solutions of two unrelated PDE solve an 'intermediate' equation?
$U = 2x^2-y^2$ and $u = x^2-2y^2$ solve constant-coefficient elliptic equations, but $U - u = x^2 + y^2$ has an interior minimum and thus cannot solve an elliptic equation.
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