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Results tagged with ap.analysis-of-pdes
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user 3709
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
9
votes
2
answers
607
views
Hyperbolic PDE in mathematics
Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most fu …
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1
vote
0
answers
79
views
Global interior estimate complex Monge-Ampere equation
Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ be a …
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0
votes
1
answer
96
views
$L^p$ regularity for semidisc
Suppose $B_r\subset \mathbb{R}^2$ is a hemidisc, i.e., $x^2+y^2 \leq r^2, y\geq 0$. Is there a regularity result of the type $\Vert \psi \Vert_{W^{2,p}(B_{1/2})} \leq C (\Vert \psi \Vert_{L^p(B_{1})} …
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12
votes
2
answers
541
views
Relationship between the Radon transform and Twistor spaces
I have often heard that the theory of Twistor spaces is ``a complex analogue" of the Radon transform. What is the precise connection ?
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2
votes
1
answer
461
views
A priori estimates for elliptic systems with bounded coefficients
Suppose we have a smooth solution to a system of PDE (for $\vec{u}$) given by $L\vec{u} = \vec{f}$ on the unit ball (no boundary conditions given). Assume that the coefficients and $\vec{f}$ are smoot …
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5
votes
1
answer
212
views
$L^p$ stability of the Beltrami equation
Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} …
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12
votes
1
answer
3k
views
Density of smooth functions in Sobolev spaces on manifolds
Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have bee …
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8
votes
1
answer
3k
views
Alexandrov-Bakelmann-Pucci maximum principle
The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient regular …
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2
votes
1
answer
496
views
Hölder estimates for the Complex Monge-Ampere equation
If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{k+2, …
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2
votes
2
answers
789
views
Monge Ampere equations (concavity)
The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\m …
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3
votes
1
answer
461
views
Regarding Discrete Eigenvalues
For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete.
But, supp …
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14
votes
1
answer
1k
views
When is a given matrix of two forms a curvature form?
Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $ …
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6
votes
0
answers
428
views
A non-elliptic PDE
I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X :
$\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial …
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1
vote
1
answer
309
views
References for weak ellipticity
There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as t …
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