This paper deals with some semilinear equations (especially Laplace equations and divergence-type equations) in the unit ball $B_1$ or the unit cylinder in the parabolic case. The author discusses local regularity of solutions; namely, all the statements about regularity are uniform in the half-ball $B_{1/2}$. Regularity of level sets of solutions and some related quasilinear problems are also investigated.

The problem studied in this paper is similar to the one studies in our first paper in the series, however, here we are solely concerned with $L^p$ estimates. For $\varepsilon>0 $ let $ u_\varepsilon $ be the solution to the following Dirichlet problem $$ -\nabla \cdot A_\varepsilon(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x,x/ \varepsilon) \text{ on } \partial D, $$ where $D \subset \mathbb{R}^d $ $ (d\geq 2) $ is a bounded domain, $g(x, \cdot)$ is $ \mathbb{Z}^d $-periodic for any $ x\in \partial D$ and the operator is uniformly elliptic. We study the problem under strict convexity of the domain $D$ and smoothness of the data involved in the problem. Our results here are twofold. First, in the case when the coefficients $A_\varepsilon(x)$ do not depend on $ \varepsilon>0 $ (i.e. for some fixed $A$ we have $ A_\varepsilon \equiv A $, which means that the operator is fixed, but not necessarily constant) then for any $ 1\leq p < \infty $ we have the following convergence result $$ \tag{1} \qquad \|u_\varepsilon - u_0\|_{L^p(D)} \leq C_p \begin{cases} \varepsilon^{1/2p} ,& d=2 , \\ (\varepsilon |\ln \varepsilon |)^{1/ p}, & d=3 , \\ \varepsilon^{1/ p} , & d \geq 4, \end{cases} $$ where $ u_0 $ satisfies the same elliptic system but with Dirichlet data set to $ \overline{g}(x):= \int_{\mathbb{T}^d} g(x, y) dy $, where $ \mathbb{T}^d $ is the unit torus of $\mathbb{R}^d $. By establishing a certain type of ergodic theorem for scaled surfaces (which essentially states that large scalings of a strictly convex smooth surface modulo $\mathbb{Z}^d$ become equidistributed in $\mathbb{T}^d$ when the size of the scaling tends to infinity) we also show that the convergence rates given in $(1)$ are (generically) sharp in dimensions higher than three, and is sharp up to logarithmic factor in dimension three.

Second, when $ A_\varepsilon(x) \equiv A(x/ \varepsilon) $ for some $ \mathbb{Z}^d $-periodic tensor $A$, (the case of simultaneous oscillations in the operator and the boundary data) combining our method with a recent result by Kenig, Lin, and Shen we prove in dimensions greater than two and for some special class of coefficients $A$, that $$ \tag{2} \qquad \|u_\varepsilon - u_0\|_{L^p(D)} \leq C_p [ \varepsilon (\ln(1/ \varepsilon))^2 ]^{1/ p }, $$ for any $ 1\leq p< \infty $. In this special case our result settles the homogenization in its optimal form. Here $u_0 $ solves the homogenized problem with constant coefficient system and boundary data (which in our special case can be computed explicitly!) depending both on coefficients of the original as well as homogenized operators, the original boundary data, and the normal field of the boundary of the domain.

The motivation for revisiting the $L^p$-estimates obtained in the first paper of the series as a corollary to pointwise bounds, comes from the following: for the problem with simultaneously oscillating operator and boundary data, as in estimate $(2)$ above, the well-known paper by Gérard-Varet and Masmoudi proves that for dimension $d\geq 2$ one has homogenization of the $\varepsilon$-problem with convergence rate in $L^2$ of order $\varepsilon^{ (d-1)/(3d+5) -\delta } $ with arbitrary small loss $\delta>0$. Comparing this result with our $L^2$-estimate from [4], in the case of constant coefficient operators when the settings of both papers coincide, we see that the convergence exponent $1/4$, which was due to [4], gives a better convergence rate than $\frac{d-1}{3d+5} $ in dimensions up to $8$, and becomes worse for $d\geq 10$. This fact served as the motivating factor for us to revised the problem and look for optimal exponents of the convergence.

The proof of $(1)$, as in [4], is based on the analysis of singular oscillatory integrals. The reason, we are able to get better estimates here, is, very roughly, due to the fact that here we are dealing with domain integration (as opposed to pointwise estimates of [4]), where the singularity of type $ |x|^{-\alpha} $ (stemming from the Poisson kernel) near the origin has a better threshold for integration.

The refined analysis of this paper gives optimal bounds in $L^p$ however does not provide any pointwise estimates.

In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla {\mathbf u}|^2+2| {\mathbf u}|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta {\mathbf u}=\frac{ {\mathbf u}}{| {\mathbf u}|}\chi_{\left\lbrace | {\mathbf u}|>0\right\rbrace}, \qquad {\mathbf u} = (u_1, \cdots, u_m) \ . $$ Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

In the previous two papers of the series we studied the homogenization problem when the domain in question was strictly convex, i.e. when the boundary was "curved" in all directions. This paper studies the other extreme, when the boundary of the domain consists of flat pieces. More precisely, we let $D\subset \mathbb{R}^d$ $(d\geq 2)$ be a bounded and convex polygonal domain, i.e. a bounded convex domain which is an intersection of finite number of halfspaces. Then for a $\mathbb{Z}^d$-periodic function $g$ and strictly elliptic matrix \(A\) we study homogenization of the following Dirichlet problem $$-\nabla \cdot A(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x/ \varepsilon) \text{ on } \partial D,$$ where $\varepsilon>0$ is a small parameter. The analysis is carried out under a certain Diophantine condition on the normal vectors of the bounding halfspaces of $D$. Namely, if $\nu \in \mathbb{R}^d$ is the normal vector of arbitrary facet of $D$, then we assume that there are constants $C, \tau>0$ such that $$ |\nu \cdot m | > \frac{C}{|| m ||^\tau}, \qquad \text{for all } m\in \mathbb{Z}^d \setminus \{0\} .$$ It is well-known and is easy to see that for any $\tau> d-1$ almost all vectors are Diophantine (i.e. satisfy the above condition) with some constant $C>0$, so the polygon is in a general position. Then, under some regularity assumptions on $A$ and $g$ we prove homogenization results for the aforementioned problem, in the form of pointwise and $L^p$-estimates, where $1\leq p< \infty$. We also discuss the sharpness of the obtained $L^p$ convergence rates depending on the geometry of the domain.

The departing point in the proofs is the integral representation of solutions via the Poisson kernel. However, due to the radical change of the geometry of the domain, proofs are quite different and difficulties lie in totally different aspects. To see how the number-theoretical condition on normals is being utilized, observe that in integral representation of solutions, through the Poisson kernel, one is lead to study quantities of the form $\int_\Pi g( y/ \varepsilon ) d\sigma(y)$, where $\Pi$ is some nice subset of a facet of $D$. Now due to the periodicity of $g$, taking $\varepsilon \to 0$ amounts to considering $\frac{1}{\varepsilon} \Pi$ modulo $\mathbb{Z}^d$. Thus, in order to preserve the ergodicity of the system, i.e. being able to recover the entire information on $g$ from the slices $\frac{1}{\varepsilon} \Pi \mod \mathbb{Z}^d$, we need $\frac{1}{\varepsilon} \Pi$ to foliate the unit cell of periodicity of $g$ as $\varepsilon \to 0$. For the foliation it is enough to have irrationality of the normal direction of the facet of $D$ containing $\Pi$, and the Diophantine condition allows one to get quantitative control over this foliations, which are needed for effective estimates for the homogenization problem. The technicalities here are handled easily using Fourier analysis.

A significant portion of the paper is devoted to establishing a certain Hölder regularity for the Poisson kernel. Here the difficulties are introduced by the corners of the polygon, where the boundary is not regular. Since the Poisson kernel is the co-normal derivative of the Green's kernel, it suffices to understand the behavior of the Green's kernel in this corners. The high-level idealogy here stems from the fact that non-negative harmonic functions in cone-like domains (with the vertex at the origin) and vanishing on the boundary, have growth controlled by the first eigenvalue of the Laplace-Beltrami operator of the surface obtained as the intersection of the cone with the unit sphere (see this paper by Ancona). With this in mind we use freezing coefficient techniques and a certain type of Phragmen-Lindelöf argument to estimate the decay of the Green's kernel in the corners. Arguing by contradiction, if the Green's kernel doesn't have the decay rate in the corners dictated by the opening of the corner, after suitable blow-ups we generate a cone, and a positive harmonic function inside it having 0 Dirichlet data on the boundary, and with a certain estimate on its growth at infinity. By comparing this harmonic function with some two-dimensional barriers (built via the bounding hyperplanes of the cone) we arrive at a contradiction, since the growth becomes less than was expected. In the course of this comparison it is important to rule out cylindrical directions of the cone, so that the cone will have a single vertex. This is being achieved by homogeneity properties of harmonic functions in NTA-conical domains (see this paper by Kuran). These results on the regularity of kernels may be of independent interest.

It should also be remarked that here our analysis is bound to scalar equations and not systems as opposed to [4] or [5].