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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.
1
vote
Liouville type theorems; linear PDE with decaying potential
For some $C$ satisfying the decay properties, such solutions do exist. Let $v$ be the function which is the fundamental solution $|x|^{2-n}$ in $\mathbb{R}^n - B_1$ (with $n \geq 3$) and $1$ in $B_1$, …
- 4,899
4
votes
Accepted
A priori estimates for elliptic systems with bounded coefficients
In general, estimates for elliptic systems with bounded coefficients are only slightly better than the energy class that solutions live in. Consider the case
$$L({\bf u}) = \text{div}(A(x)D{\bf u}) = …
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2
votes
Question About Harmonic Function Theory
This is actually pretty cool. Superharmonic functions bounded below in $\mathbb{R}^2$ are constant, while there are nonconstant superharmonic functions bounded below in $\mathbb{R}^n$ for $n \geq 3$. …
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1
vote
analysis question related to $L^p$ type inequalities
Yes, I think it's true. Say we follow the line $(x,\alpha x)$ for $x > 0$ and $0 < \alpha < 1$. Both sides of the desired inequality have no linear part at $0$, so we examine the second derivatives. K …
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7
votes
Accepted
A Liouville theorem involving an advection term
If $\Delta u + b(x) \cdot \nabla u = 0$ and $u$ is bounded, then $u$ is constant provided $|b| = O(1/(1+|x|))$. This follows from scaling, the Harnack inequality and the maximum principle.
Indeed, ad …
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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 …
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6
votes
improvement of flatness in the regularity of minimal surfaces
To show 1), one can use the idea of "calibration". Given a smooth graphical subsolution to the minimal surface equation over say $B_1 \subset \mathbb{R}^n$, extend the upward unit normal $\nu$ to $B_1 …
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18
votes
Accepted
Alexandrov-Bakelmann-Pucci maximum principle
In my view, the key point the ABP estimate is that it ties pointwise information (the PDE) to information in measure (the contact set). This is crucial to regularity theory for non-divergence equation …
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2
votes
Accepted
Regularity of the right hand side (the source term) in Evans-Krylov theory
The equation $F(D^2u) = g$ for $g \in C^{\alpha}$ should have a $C^{2,\alpha}$ estimate by perturbation theory. See for instance Caffarelli-Cabre, Ch. 8. The idea is that the constant-coefficient equa …
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6
votes
Accepted
Sobolev Inequality
This inequality cannot hold. Here is a counterexample:
Let $\phi_{\epsilon}$ be a bump function supported on $B_{\epsilon}(0) \subset \mathbb{R}^n$ with value $1$ at $0$ and satisfying $|\nabla \phi_ …
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2
votes
Dirichlet energy of a harmonic function bounded above by the energy of the boundary function?
As suggested in the very nice answer of Prof. Eremenko, I think the key point of this kind of inequality is homogeneity, and it is related to some monotonicity formulae for harmonic functions. We expa …
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1
vote
PDEs as a tool in other domains in mathematics
Reilly used PDEs to give a very elegant proof that spheres are the only embedded hypersurfaces of constant mean curvature in $\mathbb{R}^n$.
Let $\Sigma$ be such a hypersurface, bounding a region $\ …
6
votes
Accepted
A boundary Schauder estimate
One approach is to observe that
$$\|u_0\|_{L^{\infty}(B_1^+)} \geq \frac{1}{16n}|f(0)|.$$
It suffices by linearity to prove this when $f(0) = 1$. Consider the barrier
$$b(x) = \frac{1}{2n}\left(\left| …
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5
votes
Accepted
Maximum principle for heat equation, low regularity case
This version is still true: if $u$ had a local maximum at $(x,\,T)$, say with $u(x,\,T) = 0$, then $u \leq 0$ in a small parabolic cylinder centered at $(x,\,T)$. After rescaling we can assume that $u …
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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 …
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