arXiv subjects

Lidiane S. M. Lima

Publications and source records attributed to Lidiane S. M. Lima.

4 recordsLinked to original sources

arXiv2021

Adams' trace principle on Morrey-Lorentz spaces over $β$-Hausdorff dimensional surfaces

In this paper we strengthen to Morrey-Lorentz spaces the famous trace principle introduced by Adams. More precisely, we show that Riesz potential $I_α$ is continuous \begin{equation} \Vert I_αf\Vert_{\mathcal{M}_{q, \infty}^{λ_{\ast}}(dμ)}\lesssim \Arrowvertμ\Arrowvert_β^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^λ(dν)}\nonumber\\[0.02in] \end{equation} if and only if the Radon measure $dμ$ supported in $Ω\subset \mathbb{R}^n$ is controlled by $$\Arrowvertμ\Arrowvert_β=\sup_{x\in\mathbb{R}^n,\,r>0}r^{-β}μ(B(x,r))<\infty$$ provided that $1<p<q<\infty$ satisfies $n-αp<β\leq n,\; α=\frac{n}λ-\fracβ{λ_\ast}\; \text{ and }\;\frac{λ_\ast}{q}\leq \fracλ{p}\nonumber\,$. Our result provide a new class of functions spaces which is larger than previous ones, since we have strict continuous inclusions $\dot{B}_{p,\infty}^{s}\hookrightarrow L^{λ, \infty}\hookrightarrow \mathcal{M}_{p}^λ\hookrightarrow\mathcal{M}_{p, \infty}^λ \nonumber $ as $1<p<λ<\infty$ and $s\in\mathbb{R}$ satisfies $\frac{1}{p}-\frac{s}{n}=\frac{1}λ$. If $dμ$ is concentrated on $\partial\mathbb{R}^n_+$, as a byproduct we get Sobolev-Morrey trace inequality on half-spaces $\mathbb{R}^n_+$ which recovers the well-known Sobolev-trace inequality in $L^p(\mathbb{R}^n_+)$. Also, by a suitable analysis on non-doubling Caderón-Zygmund decomposition we show that \begin{equation} \Vert M_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\,\sim\, \Vert I_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\nonumber \end{equation} provided that $μ(B_r(x))\sim r^β$ on support $\text{spt}(μ)$ and $n-α<β\leq n$ with $0<α<n$. This result extends the previous ones.

math.AP↗

arXiv2021

Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-1}u+f \; & \text{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $ \mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t>0\}$ for $n\geq 2$. More precisely, inspired at \cite{deAlmeida1} and \cite{Quittner} we introduce a new functional space based in weak-Morrey spaces and we shown existence of positive solutions $u$ to the above problem when inhomogeneous term $f\in\text{weak-}\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and potential $V\in \text{week-}\mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ are sufficiently small in the natural $n/(n-1)<ρ<\infty$. Our theorems recover the range $(n+1)/(n-1)\leq ρ<\infty$ and immediately imply in solvability of the equivalent nonlocal half-Laplacian problem $(-Δ)^{{1}/{2}}u=Vu+b\vert u\vert^{ρ-1}u+ f (x)$ for $f$ and potential $V$ rough than previous ones, in view of strictly inclusions $L^λ\varsubsetneq\mathcal{M}^λ_{p} \varsubsetneq \text{week-}\mathcal{M}^λ_{p}$ for $1<p<λ<\infty$. Also, from Campanato's lemma we conclude that $u\in C^{0,α}_{loc}( \overline{\mathbb{R}^{n+1}_+})$ is locally Hölder continuous, for $f\in\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and $V\in \mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ in Morrey spaces.

math.AP↗

arXiv2014

Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burguer's equations, 2D vorticity equation, 2D generalized SQG, 3D magneto geostrophic equations, among others.

math.AP↗

arXiv2014

Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear in IUMJ (2014)] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{γ-β}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

math.AP↗