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Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size. 

Abstract We introduce a distributional Jacobian determinant $detD{V}_{\beta }\left(Dv\right)$ \det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient $V_{\beta }\left(Dv\right)={\left|Dv\right|}^{\beta }(v_{x_1},-v_{x_2})$ V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any $\beta >-1$ \beta>-1, whenever $v\in W_{\mathrm{loc}}^{1,2}$ v\in W^{1\smash{,}2}_{\mathrm{loc}}and $\beta {\left|Dv\right|}^{1+\beta }\in W_{\mathrm{loc}}^{1,2}$ \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when $\beta \ne 0$ \beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant $detD{V}_{\beta }\left(Du\right)$ \det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with $\beta =0$ \beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole ${\mathbb{R}}^2$ \mathbb{R}^{2}with $\underset{R\to \mathrm{\infty }}{lim\; inf}\underset{c\in \mathbb{R}}{inf}\frac{1}{R}{⨍}_{B(0,R)}\left|u\left(x\right)-c\right|dx<\mathrm{\infty },$ \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then $u=b+a\cdot x$ u=b+a\cdot xfor some $b\in \mathbb{R}$ b\in\mathbb{R}and $a\in {\mathbb{R}}^2$ a\in\mathbb{R}^{2}.Denoting by $u_p$ u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that $detD{V}_{\beta }\left(D{u}_p\right)\to detD{V}_{\beta }\left(Du\right)$ \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as $p\to \mathrm{\infty }$ p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that $V_{\beta }\left(D{u}_p\right)$ V_{\beta}(Du_{p})is always quasiregular mappings, but $V_{\beta }\left(Du\right)$ V_{\beta}(Du)is not in general.