In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.
We prove the validity of the $\varepsilon-\varepsilon^β$ property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.
We establish several boundary $\varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the Dirichlet boundary condition on the flat boundary. Our proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur [18], Choi-Vasseur [2], and Phuc-Guevara [6].
In this paper, we continue our work in [15] to derive ε-regularity criteria at one scale without pressure for suitable weak solutions to the Navier-Stokes equations. We establish a $\varepsilon$-regularity criterion below of suitable weak solutions, for any $δ>0$, $$\iint_{Q(1)}|u|^{\frac{5}{2}+δ}dxdt\leq \varepsilon.$$ As an application, we extend the previous corresponding results concerning the improvement of the classical Caffarelli--Kohn--Nirenberg theorem by a logarithmic factor.
We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called $\varepsilon$-scale flatness condition, which could be arbitrarily rough below $\varepsilon$-scale. This particularly generalizes Kenig and Prange's work in [32] and [33] by a quantitative approach.
In this article we are interested in studying regularity up to the boundary for one-phase singularly perturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely $$ F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = ζ_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad Ω\subset \R^n $$ where $ζ_{\varepsilon}$ behaves asymptotically as the Dirac measure $δ_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.
In this paper, we establish $\varepsilon$-regularity criteria at one scale for suitable weak solutions to the five dimensional stationary incompressible Navier-Stokes equations in both the unit ball $B_1$ and the unit half ball $B_1^+$, respectively, which improve previous results in [J. Math. Fluid Mech., 6(2004), pp 78--101] and [Comm. Pure Appl. Math., 41(1988), pp 437--458]. The proofs of the main results are based on an iteration argument, Campanato's method, and interpolation techniques.
We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on $C^2$ uniformly convex domains are $C^{1,α}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.
In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\varepsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any $p,q\in [1,\infty]$ satisfying $1\leq 2/q+3/p <2$, there exists an absolute positive constant $\varepsilon$ such that $u\in L^{\infty}(Q(1/2))$ if $$\|u\|_{L^{p,q}(Q(1))}+\|Π\|_{L^{1 }(Q(1))}<\varepsilon.$$ This is an improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application of these $…
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.
In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-\text{div} \big(A_\varepsilon \nabla u_{\varepsilon} \big) = 0$, where $A_\varepsilon(x) = A(x, x/\varepsilon_1,\cdots, x/\varepsilon_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(\varepsilon_i)_{1\le i\le n}$ are scale parameters. We show that the $C^α$-Hölder continuity with any $α\in (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $(\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_n) \in (0,1]^n$ and particularl…
This manuscript is devoted to the long-term regularity of the 2-d Navier-Stokes-Poisson system. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the rotational part of the initial velocity is assumed to be small compared to $\varepsilon$. We then show that the lifespan of the system $T^{\varepsilon}$ satisfies $T^{\varepsilon}>\varepsilon^{-(1-\vartheta)}$, where the small constant $\vartheta$ is the size of the initial perturbation in some suitable space. The …
In this article we study the gradient flow of the Möbius energy introduced by O'Hara in 1991. We will show a fundamental $\varepsilon$-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He. Ruling out blow-ups for planar curves, we will prove that the flow transforms every planar…
We prove uniform parabolic Hölder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{E}_\varepsilon(W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{ \int_{\mathbb{R}_+^{N+1}} y^a \left( \varepsilon|\partial_t W|^2 + |\nabla W|^2 \right) dX + \int_{\mathbb{R}^N \times\{0\}} Φ(w) dx \bigg\}dt, \qquad \varepsilon \in (0,1) \] where $a \in (-1,1)$ is a fixed parameter, $\mathbb{R}_+^{N+1}$ is the upper half-space and $dX = dxdy$. As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted …
We establish new quantitative Hessian integrability estimates for viscosity supersolutions of fully nonlinear elliptic operators. As a corollary, we show that the optimal Hessian power integrability $\varepsilon = \varepsilon(λ, Λ, n)$ in the celebrated $W^{2, \varepsilon}$-regularity estimate satisfies $$\frac{ \left (1+ \frac{2}{3}\left(1- \fracλΛ \right )\right )^{n-1}}{\ln n^4} \cdot \left( \fracλΛ \right) ^{n-1} \le \varepsilon \le \frac{nλ}{(n-1)Λ+λ}, $$ where $n\ge 3$ is the dimension and $0< λ< Λ$ are the ellipticity constants. In particular, $\left( \fracΛλ \right) ^{n-1} \varepsilon(…
In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivière's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.
For a parabolic equation associated to a uniformly elliptic operator, we obtain a $W^{3, \varepsilon}$ estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic $W^{2,\varepsilon}$ estimates with a comparison principle argument. As an application, we show, assuming the operator is $C^1$, that a viscosity solution is $C^{2,α}$ on the complement of a closed set of Hausdorff dimension $\varepsilon$ less than that of the ambient space, where the constant $\varepsilon>0$ depends only on the dime…
In this paper, we investigate regularity criterion for the solution of the nematic liquid crystal flows in dimension three and two. We prove the solution $(u,d)$ is smooth up to time $T$ provided that there exists a positive constant $\varepsilon_{0}>0$ such that (i) for n=3, |(u,\nabla d)|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}, and (ii) for $n=2$, |\nabla d|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}.
For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or f…
In this paper, we establish some $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$ \limsup\limits_{\varrho\rightarrow0} \varrho^{1-\frac{2}{p}-\sum\limits^{3}_{j=1}\frac{1}{q_{j}}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \leq\varepsilon, ~~\frac{2}{p}+\sum\limits^{3}_{j=1}\frac{1}{q_{j}} \leq2~~~~~\text{with}~q_{j} > 1;\\$$$$ \sup_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}} <2\quad \text{with}\, 1<q_{j}<\infty;$$ $$\…
We consider a class of equations in divergence form with a singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. $$ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}(…
We consider a 2D smectics model \begin{equation*} E_{ε}\left( u\right) =\frac{1}{2}\int_Ω\frac{1}{\varepsilon }\left( u_{z}-\frac{1% }{2}u_{x}^{2}\right) ^{2}+\varepsilon \left( u_{xx}\right) ^{2}dx\,dz. \end{equation*} For $\varepsilon _{n}\rightarrow 0$ and a sequence $\left\{ u_{n}\right\} $ with bounded energies $E_{\varepsilon _{n}}\left(u_{n}\right) ,$ we prove compactness of $\{\partial_z u_{n}\}$ in $L^{2}$ and $\{\partial_x u_n\}$ in $L^q$ for any $1\leq q<p$ under the additional assumption $\| \partial_x u_{n}\| _{L^{p }}\leq C$ for some $p>6$. We also prove a sharp lower bound on $E…
We are interested in the long time behavior of solutions of the nonlinear Schr{ö}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with $s_0>d/2$. We prove that, even in this context of low regularity, the $H^s$-norms, $s\geq 0$, remain under control during times, $T_\varepsilon= \exp \big(-\frac{|\log\varepsilon|^2}{4\log|\log\varepsilon|} \big)$, exponential with respect to the initial size of the initial datum in $H^{s_0}$, $\|u(0)\|_{H^{s_0}}=\varepsilon$. For this, we add to the linear part of the equati…
We first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak solutions. Then, an $\varepsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\varepsilon$-regularity criterion with a new approach using a local version of Leray projection operator. As an application of the approach, we obtain the short-time regularity result on a bounded domain for dissipat…
We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. A…
We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $Ω$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(Ω))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r,\,q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.
In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-\varepsil…
In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories $$\left\{\begin{array}{lll} u_{tt}- u_{xx} +bu + \varepsilon v + f(t,x,u) =0,\; v_{tt}- v_{xx} +bv + \varepsilon u + g(t,x,v) =0 \end{array}\right. $$ where $u,v$ satisfy the Dirichlet boundary conditions on spatial interval $[0, π]$, $b>0$ and $f$, $g$ are $2π$-periodic in $t$. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as $\varepsilon$ goes to 0. Firstly, under some superlinear growth an…
An $\eps$-regularity criterion at one scale for 3-D Navier-Stokes equations and improved upper bound on the box dimension of the interior singular set
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University（河北大学数学与信息科学学院机器学习与计算智能河北省重点实验室）
In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].
We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\v…
We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension $n-1$ which is also $\varepsilon$-flat in space, for any $\varepsilon>0$.
We prove that if $u\in C^0(B_1)$ satisfies $F(x,D^2u) \le 0$ in $B_1\subset \mathbb{R}^2$, in the viscosity sense, for some fully nonlinear $(λ, Λ)$-elliptic operator, then $u \in W^{2,\varepsilon}(B_{1/2})$, with appropriate estimates, for a sharp exponent $ \varepsilon = \varepsilon(λ, Λ)$ verifying $$ \frac{1.629}{\fracΛλ + 1} < \varepsilon(λ, Λ) \le \frac{2}{\fracΛλ + 1}, $$ uniformly as $\fracλΛ \to 0$. This is closely related to the Armstrong-Silvestre-Smart conjecture, raised in [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184], where the upper bound is postulated to be the optimal …
We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to 1/2 + N, as it involves nonlocal compatibility conditions.
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -Δu = f & \text{in} Ω u= 0 & \text{on} \partial Ω$$ where $f$ belongs to some $L^p(Ω)$ and $Ω$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $α\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally Hölder continuous provided that $Ω$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.
In this article, we consider and analyse a small variant of a functional originally introduced in \cite{BLS,LS} to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $\varepsilon>0$ and resembles the (scalar) Ginzburg-Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as $\varepsilon\to0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem an…
The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable versions of Gargliardo's seminorms.
We consider the 3D smectic energy $$\mathcal{E}_{ε}\left( u\right) =\frac{1}{2}\int_{Ω}\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+( u_y)^{2}}{2}\right) ^{2}+\varepsilon \left( u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$ The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on $\mathcal{E}_{\varepsilon}$ as $\varepsilon \to 0$ by introducing 3D analogues of the Jin-Kohn entropies. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature onl…
We establish sharp geometric $C^{1+α}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.
We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p-Poisson equation, for the porous medium equation and for the doubly nonlinear equation.
We obtain a sufficient condition for boundary regularity of quasiminimizers of the p-energy integral in terms of a Wiener type sum of power type. The exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant. We also show by an example that the exponent is sharp in a certain sense.
We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only on ellipticity constants and dimension, such that if $u$ is a viscosity solution of $F(D^2u) = f(x) \in L^p$, then $u\in W^{1+\varepsilon,p}$, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that $F(D^2u) \in L^p \implies (-Δ)^θu \in L^p$, for a universal cons…
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian;…
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
We discuss the sharp interface limit of a diffuse interface model for a coupled Cahn-Hilliard--Darcy system that models tumor growth when a certain parameter $\varepsilon>0$, related to the interface thickness, tends to zero. In particular, we prove that weak solutions to the related initial boundary value problem tend to varifold solutions of a corresponding sharp interface model when $\varepsilon$ goes to zero.
We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form \begin{equation}\label{Meq}\tag{Eq} u_t- F(D^2u, Du, X, t) = f(X,t) \quad \mbox{in} \quad Q_1, \end{equation} where $F$ is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $κ(n, p, q):=\frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of \eqref{Meq} belongs. We prove that when $1< κ(n,p,q) < 2-ε_F$, solutions are parabolic-Hölder continuous for a sharp, quantitative exponent $0< α(n,p,q) < 1$. Precisely at the critical border…
In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzamnn equation with Debye-Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of distribution which contains the dual of the Sobolev spaces, there exists a unique solution which belongs to a more regular Sobolev space for any positive time. We also study the sharp regularity of the solution.
