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bio website math.vt.edu/people/renardym
location Blacksburg, VA
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visits member for 4 years, 1 month
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1d
answered Companion to theoretical physics for working mathematicians
1d
comment Calculus of variation
@Figueroa: Since distributions can only be approximated by $C^1$ functions, the minimum on $C^1$ functions does not exist; it is only an infimum.
2d
answered Calculus of variation
Dec
20
comment Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Using the fact that u is harmonic, you convert $\int_\Omega \nabla u\nabla v$ to $\int_{\partial\Omega}{\partial u\over\partial n}v.$ You then end up with $z_t=Az$, where $A$ is the Dirichlet-to-Neumann map (well studied in the literature).
Dec
20
comment Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)
Your equation implies that u is harmonic, and with this restriction the trace map is invertible.
Dec
9
comment $\lVert u\rVert_{W^{2,p}}$ is bounded above by $\lVert \Delta_p u\rVert_{L^2}$ for $u \in W^{1,p}_0 \cap W^{2,p}$?
Have you looked at the one-dimensional case? Doing so should answer your question.
Dec
9
comment Solution of a second order nonlinear ode
What are you hoping for? A miracle?
Dec
9
comment How to prove it is uniformly bounded?
Since you impose the condition of zero integral, the problem at $\theta=0$ disappears. It is matter of using standard results in perturbation theory. Kato's book is a good reference.
Dec
8
comment How to prove it is uniformly bounded?
You mean for $\Omega$ to be bounded? If so, more than this is true. You get bounds not just for $\|u\|_\infty$, but actually for $\|u\|_{2,\alpha}$. Look up Schauder estimates in any textbook on elliptic PDEs.
Dec
7
answered Estimates on a heat process with fixed boundary data and zero initial conditions
Dec
4
answered $Ax=b$ in a function space
Dec
4
revised minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $
I assume you want f to be nonzero, otherwise there is an easy answer.
Dec
3
awarded  ap.analysis-of-pdes
Dec
3
comment Well-known or prolific mathematicians that have never written a sole-author article?
This question does not quite seem to fit this forum and may be closed. But as far as the impact on tenure is concerned, it is of some relevance who the coauthors are. If there are a fair number of them, you can put a positive spin on it. On the other hand, if it is mostly your colleague's thesis advisor, there may be a problem.
Dec
3
comment Inverse problem to solve out current in the radiation problem
I presume k is fixed in this? Since f has support on [-a,a], its Fourier transform is an entire function. And your formula for F gives the Fourier transform of f on the interval [-k,k]. Since an entire function is uniquely determined by its values on an interval, the inverse problem has a unique solution. But of course, analytic continuation is a highly ill-posed problem. On the other hand, most inverse problems are ill-posed. There is an extensive literature on inverse problems, and the one given here looks fairly typical. Have you made any effort to research the literature?
Dec
2
awarded  Nice Answer
Dec
2
answered If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?
Dec
1
reviewed Approve Algebraic dependency over $\mathbb{F}_{2}$
Nov
29
comment Analytical solution of diffusion PDE with Robin boundary condition
Have you tried separation of variables?
Nov
23
comment Eigenvalue problem
Your equation becomes hypergeometric under the substitution $u=\exp(-\alpha z)$.