All Questions
Tagged with ap.analysis-of-pdes fa.functional-analysis
1,304 questions
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
2
votes
0
answers
126
views
Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
2
votes
0
answers
87
views
Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $
Consider the following advection-diffusion equation
$$
\begin{cases}
u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\
u^\epsilon(0,\cdot) = u_0,
\end{cases}
$$
How can one prove an ...
- 839
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
2
votes
0
answers
71
views
Example of BV vector field $c$ without bounded divergence such that $u$ is bounded where $u_t + div(cu) = 0$
What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a ...
- 839
2
votes
0
answers
61
views
Uniqueness of solution to Cauchy problem with quadratic nonlinearity
Consider the non-linear differential operator
$$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$
For $U\subset\...
- 51
2
votes
0
answers
172
views
Can Schauder's fixed point theorem apply to a metric space?
I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
- 1,399
2
votes
0
answers
89
views
Wave equation for smooth Schwartz kernels
Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...
- 1,903
2
votes
0
answers
145
views
Integral estimate for the solution of the heat equation
Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
user140746
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
2
votes
0
answers
76
views
Weak Hessian for solutions to certain quasilinear elliptic PDEs
In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form
$$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla ...
- 3,146
2
votes
0
answers
159
views
Principal symbol of a non-local operator and Atiyah–Singer index formula
I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
- 41
2
votes
0
answers
181
views
Some questions about convergence
I'm getting lost in the first couple of sentences of a paper by Cafarelli & Alt. The sections are organized as follows, 1.1 Data, 1.2 The problem, 1.3 the statement and relevant part of proof.
1.1 ...
- 427
2
votes
0
answers
62
views
Existence and uniqueness for semilinear problem
Consider the following problem:
$$-\Delta u + [(u)^+]^\alpha = 0,$$
where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user139844
2
votes
0
answers
445
views
Lax Milgram for non coercive problem?
I obtained the variational form of my problem. and the bilinear form is below.
Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have
$$a(u,v)=\int_\Omega u(t)...
- 55
2
votes
0
answers
95
views
Exp-decay estimate of Schrodinger equation
Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
- 1,915
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
2
votes
0
answers
61
views
Second derivative of a functional defined by an integral
I was reading the following example from the book Methods in Nonlinear Analysis (Zhang, Springer) on page 10: First, everything was fine:
Example 2. Let $X = C^1(\overline \Omega, \mathbb R^N)$, $Y =...
- 21
2
votes
0
answers
148
views
Approximation of functions in $L^p(R^d;L^\infty)$
Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...
- 411
2
votes
1
answer
137
views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
user121558
2
votes
0
answers
149
views
Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
- 1,090
2
votes
0
answers
54
views
example in $L^p_{s}-$Sobolev spaces
We define $L^p-$
Sobolev spaces as follows:
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$
where $\langle \...
- 305
2
votes
0
answers
62
views
Differences among various index theories in critical point theory
Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones?
the ...
- 839
2
votes
0
answers
126
views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
- 169
2
votes
0
answers
331
views
Sobolev embeddings for vector-valued functions
I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.
In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
2
votes
0
answers
315
views
Support of a microlocal defect measure
I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
- 509
2
votes
0
answers
92
views
For what functions does Nash inequality becomes equality?
For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)?
Also same question about Poincare inequality.
- 698
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
0
answers
204
views
Eigenvalues and kernel of the the fractional laplacian in the $d$-dimensional torus
Let $\mathbb{T}^d$ be the $d$-dimensional torus. Consider the operator
$$
\Delta^{(\alpha/2)}u(x):= \int_{\mathbb{T}^d} \frac{u(x+y)+u(x-y)-2u(x)}{(d_{\mathbb{T}^d}(x,y))^{d+\alpha}}
dy$$
Where $u$ ...
- 446
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 381
2
votes
0
answers
93
views
Reference request: $|\partial_t u - \Delta u|\in L^p(\Omega^T)$ implies Holder estimate
I am currently reading a paper regarding harmonic flow between Riemannian manifolds. Let $\Omega$ be a Riemannian domain of dimension $2$, $\Omega^T=\Omega\times[0,T]$. The equation is
$$
\partial_t ...
- 1,245
2
votes
0
answers
344
views
Spectrum of Laplacian depending on boundary conditions [closed]
Consider a compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary for simplicity. Consider the Laplacian operator with zero Dirichlet boundary conditions on $\Omega$. It is well-known that $...
- 1,407
2
votes
0
answers
116
views
Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$
Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
- 1,245
2
votes
0
answers
229
views
Chain rule for Newton-derivative
I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon ...
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
2
votes
0
answers
142
views
Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
2
votes
0
answers
2k
views
Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?
In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality:
Let $\Omega\...
- 1,247
2
votes
0
answers
90
views
Boundary regularity of solutions to semilinear heat equation
Consider the Cauchy IVP problem
$$u_t - \Delta u + f(t,x,u) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Can you point out a ...
- 303
2
votes
0
answers
218
views
Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 173
2
votes
0
answers
226
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
2
votes
0
answers
115
views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
2
votes
1
answer
606
views
Strong convergence of differential quotient in $L^2(0,T;V^*)$
I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
- 187
2
votes
0
answers
140
views
Question on the differentiability of the solution mapping in the obstacle problem
I'm looking for a reference for the following. Take the obstacle problem:
$$\int_\Omega \nabla u \nabla (v-u) \leq \int_\Omega f(v-u)$$
for a function $u \in K:=\{v \in H^1_0(\Omega) : v \geq \varphi\}...
- 73
2
votes
0
answers
64
views
The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
- 679
2
votes
0
answers
235
views
The Cauchy problem associated with $u_t^\epsilon + H(x,t,u^\epsilon,\nabla u^\epsilon) = \epsilon\Delta u^\epsilon$
Consider the initial value problem $$\begin{cases} u_t^\epsilon + H(x,t,u^\epsilon,\nabla_x u^\epsilon) = \epsilon\Delta_x u^\epsilon & \text{ in } \mathbb{R}^n \times (0,\infty)\\ u^\epsilon = g &...
user81051
2
votes
0
answers
77
views
When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
2
votes
0
answers
178
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
- 157
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 21