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Results tagged with ap.analysis-of-pdes
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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Huygens' principle or finite speed of propagation?
Q: Huygens' principle or finite speed of propagation?
It's the same thing, Huygens principle is a statement of causality, which means finite speed of wave front propagation.
Without Huygens principle …
- 188k
6
votes
Accepted
Are renormalizability and the criticality of a PDE synonymous?
The terms describe how the coupling terms of the theory change as one increases the energy. A theory is renormalizable = critical if the coupling terms remain unchanged, super-renormalizable = sub-cri …
- 188k
6
votes
Why is resonance such a widespread phenomenon?
A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\o …
- 188k
5
votes
Reference request: Parabolic Equations
Back in 2012, professor Ben Chow gave some advice to a similar question; these include the
Lectures on Elliptic and Parabolic Equations in Sobolev Spaces by Krylov [recommended here by Giorgio Metafu …
- 188k
4
votes
Euler-Lagrange equations for minimizer of energy with indicator function
Edit: An earlier version of this answer missed a factor of two, now corrected, thanks to Daniele Tampieri.
We seek the variation of the functional
$$L[u]=\int_\Omega\left(|\nabla u|^2+1\right)\chi_{u> …
- 188k
6
votes
Rigorous treatment of Ostrogradsky's instability theorem?
On the problem of stability for higher-order derivative Lagrangian systems in Letters in Mathematical Physics (1987) may have the desired level of rigor (see Theorem 1).
The proof of the theorem is a …
- 188k
12
votes
Reference request: Software for producing sounds of drums of specified shapes
The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on whe …
- 188k
1
vote
Accepted
Equivalence of Wind Forces: Intensity vs. Duration
You should integrate power, not force.
Wind dissipates kinetic energy when it hits a structure. The dissipated power on an area $A$ is given by
$$P = \tfrac{1}{2} A \rho v^3,$$
where $\rho$ is the mas …
- 188k
3
votes
Accepted
Perturbation methods for stochastic/partial differential equations
An older source is
Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics (1980),
a more recent source is
Perturbation Theory for Stochastic Differential Equations …
- 188k
2
votes
Feynman–Kac formula for other operators
Some pointers to the (extensive) literature on generalized Feyman-Kac formulas:
Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian
Fractional …
- 188k
4
votes
Gradient flows and particle representations
You want to derive the Fokker-Planck equation (the drift-diffusion equation for the density) from the Langevin equation (the stochastic differential equation for the position of a particle); this is s …
- 188k
1
vote
Accepted
Interpretation of the singular integral for the definition of fractional Laplacian in classi...
One way to interpret the singular integral for $s=1-\epsilon$ is by Fourier transformation, to check that it tends to $k^2 \hat{u}(k)$ when $\epsilon\downarrow 0$.
I will make use of the fact that the …
- 188k
4
votes
Accepted
Elliptic PDEs in Finance
For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "mathematical fi …
- 188k
9
votes
Accepted
Prove J.L. Lions’s Lemma without using Fourier transform
Lion's lemma is equivalent to several other properties that have simpler proofs, see On a lemma of Jacques-Louis Lions and its relation to other fundamental results:
Some of these equivalent properti …
- 188k
6
votes
Accepted
The determinant as a differential operator
Gårding's differential operator (introduced in Extension of a Formula by Cayley to Symmetric Determinants) is discussed by Turnbull in Symmetric Determinants and the Cayley and Capelli Operator:
The …
- 188k