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 4h comment Eigenvalue inequality for regular graphs @DavidE.Roberson Yes, I would be very interested in the proof. Characterising a subclass of graphs by spectral means is often a very relevant step, in particular if you could also prove some kind of monotonicity (the "less strongly regular" a graph is, the larger the gap in your inequality). 5h comment Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$ @FanZheng You can even write never, for that matters, unless the parabolic equation is associated with a bounded linear operator. But then, I never claimed analyticity in 0 of anything :) 14h comment Regularity on Neumann problem on polygonal domain Thanks, I didn't know that paper by Dauge. I must admit I don't understand it quite precisely, but still: If one tries to apply her Theorem 1.1, don't her remarks right after the theorem yield the regularity you are looking for if and only if $p<6/(5-\sqrt{5})\simeq 2,17$ (the first condition being automatically satisfied in the case of a cube, where $\omega=\pi/2$ and $k=0$)? 1d comment Regularity on Neumann problem on polygonal domain In the case of domains smooth enough I have never seen a dependence on $p$ - even the spectrum and the eigenfunctions of all these operators is identical! But I do know instances of $p$-dependence in the case of certain nonlinear operators, like the $p$-Laplacian (but here $p$ has nothing to do with "your $p$"). So I would be interested in the kind of results you are mentioning. 1d comment Regularity on Neumann problem on polygonal domain The philosophy in that book is that if anything can go wrong in a domain with a nonsmooth domain, it is because of sharp edges/corners, which means that for convex domains the situation will generally be benign. That said, the result you are looking for is Thm. 3.2.1.3 in the case $p=2$, but I had no time to look for the general case. 2d comment Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$ Proving analyticity in time for a quasilinear parabolic problem? Good luck! :) 2d comment Regularity on Neumann problem on polygonal domain Can you please explain why the results by Grisvard are not sufficient? He also considers spaces of Hölder continuous functions btw. 2d comment heat equation in 2D with absorbing and reflecting boundary conditions @mas19 With the fourth line, the problem is overdetermined! 2d answered Compact embedding and fractional Sobolev spaces in unbounded domain Nov 19 comment A metric on the set of BV functions, is it mentioned/studied in literature? @GeraldEdgar I do not see that BV implies square integrability. To the best of my knowledge one has the imbeddings $W^{1,1}(0,1)\hookrightarrow BV(0,1)\hookrightarrow L^1(0,1)$. Does $BV(0,1)$ imbeds in anything better than $L^1(0,1)$? Nov 18 comment Can one hear the shape of a drum for operators? @bigM: I am not aware of any. Should such a hypothetical domain be non-smooth, it would be a rather interesting object in spectral geometry. Nov 18 comment Hahn-Banach theorem with convex majorant > "the result is even interesting for X=R" I cannot understand this remark. If $L=\{0\}$, then the assertion (that there exists a linear function below the convex function) is trivial, and even more so if $L=\mathbb R$. Or am I overlooking something? Nov 16 comment Under what conditions does a continuous-time Markov chain is also a Feller process? > is a continuous-time Markov chain with finite number of states are Feller? Yes. A Markov semigroup on $L^1(X)$ is also Feller if leaves invariant $C_0(X)$. But if $X$ is finite, then these both spaces agree. Nov 16 comment Famous vacuously true statements and what about the theory of $\mathbb F_1$? Nov 15 comment Can one hear the shape of a drum for operators? Let me add that, as far as I know, Kac's question is still open for smooth domains. Milnor's examples are merely Lipschitz continuous... Nov 12 comment What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem? I assume this should be community wiki. That said, there is a nice entry in Trevisan's blog about this: lucatrevisan.wordpress.com/2015/11/03/… Nov 12 comment Distance between quadratic forms (I must add I find it a bizarre notion of distance, since $d(q,q')=1$ if $q=q'$. But then I do not understand the difference between $q$ and $[q]$, so probably I am misunderstanding.) Nov 12 comment Distance between quadratic forms Take e.g. $q'(f)=\|\nabla f\|_{H^1_0(\Omega)}^2$ and $q(f)=\|f\|^2_{L^2(\Omega)}$, $\Omega$ open and bounded. What you get is the $\sup$ and the $\inf$ of the Rayleigh quotient for the Laplacian with Dirichlet b.c., respectively. But I suppose this is not the setting the author of that paper had in mind. Nov 11 comment Distance between quadratic forms Courant-Fischer Theorem. Nov 11 comment Distance between quadratic forms I don't know whether this is the motivation for this definition, but if $q'$ and $q$ are closed, positive forms on Hilbert spaces $V,H$ and $V$ is densely and continuously embedded in $H$, then the numerator (resp., denominator) of your quotient is the largest (resp., smallest) eigenvalue of the operator $A$ on $H$ associated with $q'$, so the distance will typically have value $\infty$ if $A$ is an unbounded operator.