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.
4,466 questions
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Asymptotic expansion of the solution of a nonlinear wave equation
I am reading the article Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime (arXiv link) by Barreto, Plamen, where they consider the modified cubic NLWE
$$-\...
- 890
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1
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136
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Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
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105
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KdV/KP-II equation with upper semicontinuous initial data and viscosity solutions
In the article "KP governs random growth off a 1-dimensional substrate", they study the KP-II equation: the function $\phi(t,x,r)=\partial_{r}^{2}\log(F)$ satisfies
$$\partial_{t}\phi+\frac{...
- 5,474
4
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198
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Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
- 51
4
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1
answer
223
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Poincaré inequality on compact manifolds without boundary
The question arises from a paper The heat flow for the full bosonic string, Ann. Global Anal. Geom. 50 (2016).
In line 4, Page 362, the author claimed the following inequality which looks similar to ...
- 143
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80
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Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
2
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1
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117
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Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
- 1,759
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Solving $X$ for prescribed $\operatorname{div}(X)$ of compact support
Let $f \in C^\infty_0(\mathbb{R}^3)$ be a smooth scalar function of compact support. I would like to find a smooth vector field $X : \mathbb{R}^3 \to \mathbb{R}^3$ satisfying $\operatorname{div}(X) = ...
- 403
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179
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Green function of an elliptic operator
Let $L$ be an elliptic operator on $\Bbb C$ with
$$\DeclareMathOperator{\Img}{Im}
L^{-1}f(z)=\int_{\Bbb {C}}K_1(|z-w|^2)f(w) e^{i\cdot\Img\langle z,\overline{w}\rangle} \, dw
$$ where $K_1$ is the ...
- 109
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1
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Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
- 452
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1
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152
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Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
- 221
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votes
1
answer
518
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Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
- 435
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197
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On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 1,429
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870
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Can two drums almost sound the same?
Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$).
Mark Kac,...
- 3,054
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2
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408
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Does there exist an electromagnetic analogue of Einstein's field equations?
This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything.
I recently learned about gravitoelectromagnetism which describes an ...
- 2,689
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50
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"Reverse monotonicity" for k-Hessian measures
My question deals with the $k$-Hessian measure $\mu_k[u]$ of a $k$-convex function $u\in \Phi_k(\Omega)$ as defined in the articles "Hessian measures I-III" by N. S. Trudinger and X. J. Wang....
5
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1
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140
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Dispersion of random walk with scaled step sizes
Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$.
We define the ...
- 452
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1
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142
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Existence of first variation
I am trying to compute the first variation of the functional
$$\mathcal F(\rho) = \int_{\Omega} R(x;\rho) d\rho(x)$$
where $R$ is some function of $x$ that also depends on $\rho$. Here $\rho$ is a ...
- 21
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139
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Understanding a "straightforward" application of the method of stationary phase for proving a trace formula on compact hyperbolic surfaces
I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch (Duke Mathematical Journal, 55, pp. 919-941 (1987), MR916129, Zbl 0643.58029) and am ...
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A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
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0
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167
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Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
2
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65
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Generalized Fourier transforms associated to Schroedinger operators
Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
- 4,115
2
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42
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Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
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0
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74
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Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
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5
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PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 8,998
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91
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Sobolev embedding on a compact manifold without boundary
I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996).
In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
- 311
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132
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Uniqueness for a nonlinear kinetic PDE-system with heat transfer coupling in one dimension
I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, ...
- 185
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120
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Well-posedness result for a linear parabolic equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 185
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446
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Is the uniform limit of "almost eikonal" maps eikonal?
Let $f_n: \mathbb R^d \to \mathbb R$ be continuously differentiable functions with $f_n \to f$ uniformly for some $f$.
Suppose that $|\nabla f_n| \to 1$ uniformly. Is it true that $f$ is $C^1$ with $\...
- 6,321
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1
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263
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Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 825
1
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1
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Well posedness of the Plateau problem under lack of uniqueness
The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not.
Premises
I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
- 6,367
5
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0
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105
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Convolution of a bounded function and measures
Given a function $f\in L^\infty(\mathbb{R}^n)$ and a family of Radon measure $\mu_\alpha$, under what condition do we have $f*\mu_\alpha$ equi-continuous?
One condition I know is if $\mu_\alpha$ has a ...
- 375
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92
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Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
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0
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89
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Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
- 61
1
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1
answer
165
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Calderón–Zygmund inequality for Neumann problems
My question is simple:
Let $\Omega$ be a bounded smooth domain. Does the following inequality $$\|D^2w\|_{L^q(\Omega)}\leq C\|\Delta w\|_{L^q(\Omega)},$$ hold for any function $w\in W^{2,p}(\Omega)\...
- 113
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Continuity dependence and convergence of the renormalized $\Phi^4_2$ model
This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.
Again, we are interested in the local behavior of the $\Phi_2^...
- 573
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votes
1
answer
211
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Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 825
0
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0
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42
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Fiber-wise mappings composed with projection map $\pi$
Let $M^2=(0,1)^2$. Recall that a chart is a diffeomorphism $\varphi:M^2 \to M^2$. Given a chart $\varphi:(M^2,g_0)\to (M^2,g_0)$ for $g_0$ the Euclidean metric, consider the curves $\varphi^{-1}(u,t)=\...
- 383
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Does nice behavior near a singular point force solution to be in Frobenius series?
I have a pair of partial differential operators $\Delta_1$ and $\Delta_2$ in $y_1, y_2$ formed from constants, multiplication by $y_1$ or $y_2$ and derivatives in the form $y_1 \frac{\partial}{\...
- 151
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Understanding the Schrodinger flow——Symplectic Banach manifold
This question was posted on https://math.stackexchange.com/questions/4925369/understanding-the-schrodinger-flow-symplectic-banach-manifold but recieve nothing. I really want to know the something ...
1
vote
1
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142
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Hardy inequality
Suppose $B_1$ is the unit ball centered at the origin in $ R^N$ with $N \ge 3$. Let $ q= 2^* = \frac{2N}{N-2}$. Does there exist some $C>0$ such that
$$\int_{B_1} \frac{ u(x)^2}{|x|^2} dx \le C ...
- 1,385
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Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
3
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196
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Towards Schauder estimates: smoothing effect of the semi-group generated by $\Delta+(-\Delta)^{1/2}$
Consider the semi-group $(P_r)_r$ generated by $\Delta+(-\Delta)^{1/2}:$ for a distribution $f$ let $P_rf:=p(r,\cdot)*f$ where $p(r,x):=\sum_{q \in \mathbb{Z}^d}e^{2\pi\mathrm{i}\langle q,x\rangle}e^{-...
- 573
2
votes
1
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94
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Quasilinear wave equations without (weak) null conditions and conjectures
I have found that most works on quasilinear wave equations require, at least, the (weak) null condition. There are only a few works without this condition, such as "Shock Formation in Small-Data ...
- 89
2
votes
0
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94
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Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
3
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0
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148
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Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
4
votes
3
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473
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Generalized Fuchsian-type PDE
Consider
$$
\big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0
$$
with the initial condition $A(x,0)=1$. In a small $t$...
- 141
2
votes
2
answers
151
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Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
4
votes
1
answer
211
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Local solutions of renormalized stochastic PDE
To illustrate the problem consider the mild formulation of the $\Phi^4_2$ model on $[0,T]\times \mathbb{T}^d$: $$\phi=P_r\phi_0+\int_0^rP_{r-q}(-\phi^3(q))dq+Y_r \ \ \ \ \ \ (1)$$ where $(P_r)_{r \...
- 573
0
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0
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116
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Integral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:
$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$
$f(\...
- 1,759