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Results tagged with ap.analysis-of-pdes
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user 122484
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
1
answer
202
views
Approximation of functions by tensor products
Given a 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 sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(y)$ su …
- 411
2
votes
0
answers
147
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
$$
\big …
- 411
4
votes
1
answer
271
views
Is the maximal function bounded on the Besov space?
The Hardy-Littlewood maximal function of a function $f$ is defined by
$$
M f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}|f(x+y)|dy,
$$
where $|B_r|$ denotes the Lebesgue measure of the ball $B_r$. …
- 411
2
votes
1
answer
169
views
Solving classical parabolic equation by using Littlewood-Paley theory
Consider the following classical PDE in $R^n$:
$$
\partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0.
$$
Is there any references on solving the above equation by using the L …
- 411
0
votes
0
answers
457
views
Is the product of two Sobolev functions in L^p?
Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$.
Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$.
Under what conditions on $\beta$ can we get that $fg\in …
- 411
4
votes
0
answers
191
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n …
- 411