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1d
comment A comparison principle for degenerate parabolic equation
What is the PDE, then? it seems you have time derivatives on the boundary, which I've never seen before...
1d
comment A comparison principle for degenerate parabolic equation
Is $p\in R$ really negative, or is it just a typo? In which sense do you understand $\varphi '$ (I guess $\partial \varphi/\partial t$) if you only have $\varphi\in \mathcal{C}^1(0,T;L^2(\Omega))$ as a time regularity? Of course with your assumptions you also have $\varphi\in L^{2}(0,T;H^1(\Omega))\hookrightarrow L^{2}(0,T;H^{1/2}(\partial\Omega))\hookrightarrow L^{2}(0,T;L^2(\partial\Omega))$, but this still doesn't make sense of the time derivative on the boundary. I'm a little surprised by your $\partial\Omega$ boundary terms, can you tell us what is your (degenerate) parabolic PDE?
2d
answered reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
Apr
13
asked reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
Apr
12
revised Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
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Apr
12
revised Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
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Apr
12
revised Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
added 63 characters in body
Apr
12
revised Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
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Apr
12
answered Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
Apr
9
comment Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
I wasn't arguing about the criticality, only pointing out that your function $\rho_m$ is not in $L^3(R^3)$ (the problem is integrability at the origin). Again: it is radial and behaves as $1/r$ at the origin, but $1/r\notin L^{3}$ in any neighborhood of the origin (in dimension 3). As a consequence it doesn't even make sense to look at the criticality, since you have $h(\rho_{m,R})=\infty$ for any $R>0$.
Apr
8
comment Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
you should be really careful for several reasons: 1) your function $\rho_m$ does not belong to $L^3$ (in dimension 3 with $dx\approx r^2dr$ this would require $\frac{1}{r^3}r^2dr$ to be integrable near $r=0$, which is obviously false). 2) for random $\eta$ your perturbation $\rho_m+t\eta$ may not be an admissible candidate (you may loose positivity/non-negativity, or violate the constraint $|\rho|_{L^1}=1$). So even convexity may not be enough and you have to prove your claim "by hand". I'm working on it, I believe the result holds.
Apr
6
comment Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $
I guess by stationary function you mean critical point? In which case of course you absolutely cannot conclude anything, your $\rho_m$ may be a saddle point...
Mar
30
revised A useful criterion in vector integration
Just a few typos fixed and formatting improved.
Mar
29
answered A useful criterion in vector integration
Mar
8
comment reference request: Riesz/Newton potential and HLS inequality in L1.logL1
Thank you Piero, this looks promising and I will check it out.
Mar
6
accepted uniqueness for Poisson equation in R^d with mildly regular data
Mar
6
comment uniqueness for Poisson equation in R^d with mildly regular data
Thank you Piero, that works fine!
Mar
6
asked uniqueness for Poisson equation in R^d with mildly regular data
Mar
4
comment Is this integration by parts legitimate?
OK, thank you Alexandre. What if the dimension is now $d=2$ and the Poisson Kernel $-\log|x-y|$?
Mar
4
accepted Is this integration by parts legitimate?