On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary
Jesus Idelfonso Díaz Jean Michel Rakotoson
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 1037-1058 doi: 10.3934/dcds.2010.27.1037
We prove the existence of an appropriate function (very weak solution) $u$ in the Lorentz space $L^{N',\infty}(\Omega), \ N'=\frac N{N-1}$ satisfying $Lu-Vu+g(x,u,\nabla u)=\mu$ in $\Omega$ an open bounded set of $\R^N$, and $u=0$ on $\partial\Omega$ in the sense that

$(u,L\varphi)_0-(Vu,\varphi)_0+(g(\cdot,u,\nabla u),\varphi)_0=\mu(\varphi),\quad\forall\varphi\in C^2_c(\Omega).$

The potential $V \le \lambda < \lambda_1$ is assumed to be in the weighted Lorentz space $L^{N,1}(\Omega,\delta)$, where $\delta(x)= dist(x,\partial\Omega),\ \mu\in M^1(\Omega,\delta)$, the set of weighted Radon measures containing $L^1(\Omega,\delta)$, $L$ is an elliptic linear self adjoint second order operator, and $\lambda_1$ is the first eigenvalue of $L$ with zero Dirichlet boundary conditions.
    If $\mu\in L^1(\Omega,\delta)$ we only assume that for the potential $V$ is in L1loc$(\Omega)$, $V \le \lambda<\lambda_1$. If $\mu\in M^1(\Omega,\delta^\alpha),\ \alpha\in$[$0,1[$[, then we prove that the very weak solution $|\nabla u|$ is in the Lorentz space $L^{\frac N{N-1+\alpha},\infty}(\Omega)$. We apply those results to the existence of the so called large solutions with a right hand side data in $L^1(\Omega,\delta)$. Finally, we prove some rearrangement comparison results.

keywords: Very weak solutions; semilinear elliptic equations; distance to the boundary; weighted spaces measure; unbounded potentials.
Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach
Jesus Ildefonso Díaz David Gómez-Castro Jean Michel Rakotoson Roger Temam
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 509-546 doi: 10.3934/dcds.2018023

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\text{IR}^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

keywords: Linear diffusion equations singular absorption potential unbounded convective flow no boundary conditions dual problem local Kato inequality distance to the boundary weighted spaces
Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data
Alberto Fiorenza Anna Mercaldo Jean Michel Rakotoson
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 893-906 doi: 10.3934/dcds.2002.8.893
In this paper we prove some regularity and uniqueness results for a class of nonlinear parabolic problems whose prototype is

$\partial_t u - \Delta_N u=\mu$ in $\mathcal D'(Q) $

$u=0$ on $]0,T[\times\partial \Omega$

$u(0)=u_0$ in $ \Omega,$

where $Q$ is the cylinder $Q=(0,T)\times\Omega$, $T>0$, $\Omega\subset \mathbb R^n$, $N\ge 2$, is an open bounded set having $C^2$ boundary, $\mu\in L^1(0,T;M(\Omega))$ and $u_0$ belongs to $M(\Omega)$, the space of the Radon measures in $\Omega$, or to $L^1(\Omega)$. The results are obtained in the framework of the so-called grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

keywords: Grand Sobolev spaces uniqueness measure data parabolic equations. regularity

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