Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,271
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The energy of a semilinear ODE
I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
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2
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Sharp Dirichlet heat kernel estimates in exterior domains?
I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann ...
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Convergence of spheres in nonlinear stability of Minkowski space
My question is about Lemma 3.3.1 in Christodoulou and Klainerman's proof of nonlinear stability of Minkowski space. This lemma says the following: Consider a family of metrics $m_u$ on $S^2$ defined ...
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What does it mean by "converges boundedly"?
On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
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Convergence of a level set when $f^n\to f$ in $C^1$ sense
Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that
$$f^n\to f$$
in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in ...
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Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
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Alternative proof of Liouville theorem for harmonic functions
From Prove Liouville theorem without using mean value property the following question arises:
To prove the Liouville theorem
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
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Variational formulation of an elliptic pde
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem:
$$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$
$$ V(x,...
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If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets?
Let $u_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi_\delta$ a family of standard mollifiers with compact support. Hörmander claims in The Analysis of Linear Partial ...
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Improving dispersive estimates for linear KdV
Consider the equation
$$\partial_t u(t,x) = -\partial_{xxx} u(t,x)$$
for $x \in \mathbb R.$
It is well-known that in general we have
$$\Vert u(t) \Vert_{L^{\infty}} \le C t^{-1/3} \Vert u_0 \Vert_{L^1}...
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Boundary regularity of rectifiable multiplicity 1 hypercurrents
Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
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How to find the conserved quantities of the Kirchhoff equation?
Consider the Kirchhoff equation, given by
$$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
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Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
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Regularity and normal trace of "Hdiv" measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
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Inequality for initial data
I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
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Energy Decay of the functional $\int_{B_1} |Du|^2 +Au^2$
Suppose $u \in C^1(B_1)$ with $B_1 \subset \mathbb{R}^n$ such that $\Delta u =0$ weakly. We would have the energy decay estimate
$$\int_{B_r} |Du|^2 \leq r^n \int_{B_1} |Du|^2.$$
Now suppose $u \in C^...
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regularity of p-harmonic functions
We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.
Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
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Solution singular PDE
I've been studying the following singular PDE
$$
\mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$
in $\Omega \subset \mathbb{R}^{2}$.
Do you know any reference, ...
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The Poisson equation
I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates
$$
\triangle u=f\quad in \quad \> B_2. \>\quad \quad \quad \quad (1)
$$
Lemma 7: There is a ...
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Closure of tensor product /tensor product semigroup
In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user127612
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Reconstructing the Green's function of an initial-value problem of partial differential equation
Consider a partial differential equation that is of the following form:
\begin{equation}
(-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t)
\end{equation}
where $g(x)$ is a real function. Suppose that $f(...
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Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
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Rate convergence of the heat equation as diffusion tends to zero
Is there a good reference for the following problem? Consider any smooth bounded domain $\Omega$ and solve the heat equation
\begin{align}
\partial_t u^\kappa &= \kappa \Delta u^\kappa,\\
u^\...
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votes
1
answer
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Can I approximate a function of bounded variation with orthogonal polynomial?
Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e.,
$$
u = \...
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Hadamard-Rybczynski problem
HR problem deals with a spherical fluid viscous drop falling in a different fluid under influence of gravity. The outer fluid is of uniform speed $U$ in direction of gravity, far away from the drop. ...
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Parabolic Regularity with Neumann B.C
Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)...
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Looking for access to McKean's original paper?
I'm looking for the PDF version/scan of Henry P McKean Jr.'s paper on propagation of chaos. The reference is as follows -
Propagation of chaos for a class of non-linear parabolic equations., In ...
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answers
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Relation between controllability and stability of PDE
In general, when we talk about controllability, we talk about proving the existence of a control input that transfers the state to a desired state at a desired time $T$. However, when we talk about ...
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Orthogonality to harmonic functions
Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define
$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
$b_n=f\Delta^{...
3
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Prove existence of continuous function on $(0,1)$ with special properties [closed]
Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$.
I am asking if there is a continuous function $h$ such that
$$\int_0^1 h(s) f(s) ds=0$$
$$...
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1
answer
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Semi-elliptic pdes reference
The pde
$$ Pf(x)=\sum _{i,j=1}^{n}a_{ij}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)+\sum _{i=1}^{n}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+c(x)f(x),$$
is said to be semi-...
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Does Huygens principle holds for heterogeneous media (variable coefficients)?
I'm having trouble to find references on that. Consider for instance a very simple model of a wave equation with variable coefficients:
$$\partial_{tt}^2 u(x,t) - \nabla \cdot( a(x) \nabla u(x,t)) = f(...
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Finding a smooth convex function with prescribed boundary value and small Monge-Ampère measure
Let $\Omega\subset\mathbb{R}^n$ be a bounded strictly convex domain and $\nu:\partial\Omega\rightarrow\mathbb{R}$ be a lower semi-continuous functions. It is well known that the function $\overline{\...
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Free quantum evolution operator on Sobolev space
I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
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Bounded solutions for Schrödinger equation at the edge of the essential spectrum
Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
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2
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Bound of solution of pde with a distance function
I would like to solve the PDE $\Delta u=-K$ in $\Omega$ and $u=0$ on the boundary, where $K$ is some positive constant. I read a paper which stated that $u(x)$ is no less than the distance from $x$ to ...
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The space $L^p(\partial\Omega)$ in cited references
The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
user14319
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Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE
Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for ...
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elliptic regularity of Neumann problem on Square
I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{...
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A question on the Frechet derivative
Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in W_0^{1,p}(\...
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Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
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Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$
Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
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Application of Egorov's Theorem for Pseudodifferential Operators
Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
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semi-classical Green's function
I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations.
In the two papers, he gave a semi-classical approximation of the Green's function to ...
- 577
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1
answer
370
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What are the applications of Grillakis Shatah and Strauss paper?
I am studying the following paper.
Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197.
...
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votes
1
answer
136
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$C_0$ semigroups on parameterized Banach spaces or moving domains
Is there any literature corresponding to one or two-parameter semigroups such that e.g. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$?
I ...
- 153
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votes
1
answer
939
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Strong maximum principle for weak solutions
Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
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1
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Decay of Solutions to the Heat equation
Consider the heat equation
$$ (\partial_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.
Consider the semigroup ...
- 9,591
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1
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206
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Coercivity for functional and complete orthonormal system
Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional
$$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$
I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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Elliptic Fourier integral operators
I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...
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