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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.

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        Existence and regularity for fractional Poisson-type equation

        According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results. Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
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        Harmonic oscillator in spherical coordinates

        It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
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        278 views

        Klein Gordon equation - references

        The Klein Gordon equation of the form: $\Delta u+ \lambda u^p=0$ is been studied for $p = 2$? (i.e.$\Delta u+ \lambda u^2=0$) If yes are there references?
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        geodesic balls in the conformal change

        Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...
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        What are tools we need to prove scattering result for NLS when propagator is unitary?

        Consider nonlinear Schr?dinger equation (NLS) $$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$ where $F$ is some nonlinearity. Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...
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        Global interior estimate complex Monge-Ampere equation

        Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
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        297 views

        Does current follow the path(s) of least (total) resistance?

        Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
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        62 views

        Regularity in Orlicz spaces for the Poisson equation

        I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007) $$\Delta u=f \quad \quad \quad \quad (1)$$ Lemma 2: There is a constant $N_1 >1$ so that ...
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        86 views

        Conserved Positive Charge for a PDE

        Let $(x,t) \in \mathbb{R}^2$, $W(x)$ be a (smooth enough) real-valued function and consider the following partial differential equation for the real-valued function $U(x,t)$ \begin{equation} \frac{\...
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        Elliptic pde L^p theory via adjoint theory

        Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...
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        136 views

        The Poincaré inequality for the Sobolev space on a domain with a non Lipschitz boundary

        Let $\Omega$ be a bounded open Lipschitz domain in $\mathbb{R}^{d}, d \geq 3$. Assume that $L$ is a straight segment such that $L \subset int(\Omega)$. Let $v \in V:= \overline{ \{ \phi \in C^{\infty}...
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        Weak elliptic maximum principle on manifolds without strict ellipticity

        This question is not to be confused with the similarly titled question here. In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...
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        1answer
        84 views

        Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace?

        The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not ...
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        62 views

        Wave equation with 'spring' integral boundary condition

        I am really stuck with this small toy problem. I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is: ...
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        Interpolation inequalities involving mean curvature operator

        Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator $$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$ (in any ...

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