Exterior-sphere elliptic Dirichlet existence, energy and residual stability

Expand the intended divergence expression by the product rule to obtain $L=A:D^2+\beta \cdot D+\gamma$. On a bounded Lipschitz domain, two sufficiently close points in a fixed smaller graph patch can be joined inside a larger patch by a path of length at most a fixed multiple of their distance; bounded first derivatives therefore give a Lipschitz bound for $a,b$ there. A finite cover and the supremum bound for separated pairs give their global Hölder bounds. Thus the finite global coefficient assumptions give finite Hölder bounds for the expanded coefficients, $\gamma \le 0$, and a finite bound for $DA$. Throughout the proof write $c=\gamma$, and use $\lambda ={\lambda }_0$, $\mathrm{\Lambda }={\mathrm{\Lambda }}_0$.

If $n=1$, the bounded Lipschitz set is a finite union of open intervals: its compact boundary has a finite one-sided graph-chart cover, and in dimension one each such chart contains one boundary point. On each interval $(a,b)$ all coefficients and $f$ extend Hölder-continuously to $[a,b]$ and $A\ge \lambda$. Write $y=(u,u^{\mathrm{\prime }})$, so $y^{\mathrm{\prime }}=M(x)y+F(x)$, where $M=\left(\begin{array}{cc}0& 1\\ -c\mathrm{/}A& -\beta \mathrm{/}A\end{array}\right)$ and $F=(0,f\mathrm{/}A)$. For $Th(x)={\int }_a^{x}M(t)h(t)dt$, iterated integration gives $\mathrm{\parallel }{T}^{k}h{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}(\mathrm{\parallel }M{\mathrm{\parallel }}_{\mathrm{\infty }}(b-a){)}^k\mathrm{/}k!$. Therefore the uniformly convergent series $y={\sum }_{k\ge 0}{T}^k(y(a)+{\int }_a^{\cdot }F)$ solves the integral equation, hence the differential system; the factorial bound also proves uniqueness by iterating the homogeneous difference equation. Choose one particular solution with the prescribed left value and slope zero, and add $s$ times the homogeneous solution with initial value zero and slope one. That homogeneous solution cannot also vanish at $b$: otherwise [F2] would make it identically zero, contradicting its slope at $a$. Thus exactly one $s$ gives the prescribed right value. The equation makes $u^{\mathrm{\prime }\mathrm{\prime }}$ globally Hölder, so $u\in C^{2,\alpha }([a,b])$. Zero endpoint values imply $u\in H_0^1(a,b)$ directly: multiply by a cutoff vanishing within $\epsilon$ of each endpoint, with derivative bounded by $C\mathrm{/}\epsilon$; since $\mathrm{\mid }u\mathrm{\mid }\le \mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{dist}(x,\mathrm{\partial }(a,b))$, the cutoff error in $H^1$ tends to zero, and each compactly supported $H^1$ function is approximated by mollification. For zero data, integration against $u$ gives $\lambda \mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }u{\mathrm{\parallel }}_2+\mathrm{\parallel }\beta -A^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_2\mathrm{\parallel }u{\mathrm{\parallel }}_2+\mathrm{\parallel }c{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$; Young's inequality and [F2] then give the stated energy bound on these finitely many intervals. This proves all existence, energy, and global regularity conclusions in dimension one; [F2] gives stability. Henceforth take $n\ge 2$.

Let $\delta (x)=\mathrm{dist}(x,{\mathbb{R}}^n\setminus U)$ on all of ${\mathbb{R}}^n$; it is nonnegative, 1-Lipschitz, vanishes outside $U$, and has compact support because $U$ is bounded. Fix a nonnegative unit-mass smooth mollifier supported in the unit ball. Put ${\epsilon }_j\downarrow 0$ and ${\delta }_j={\rho }_{{\epsilon }_j}\ast \delta$. Then $\mathrm{\mid }{\delta }_j-\delta \mathrm{\mid }\le {\epsilon }_j\mathrm{.}$ Choose a regular value $t_j\in (2{\epsilon }_j,3{\epsilon }_j)$, using the local Euclidean Morse–Sard theorem, and set $V_j=\{{\delta }_j>t_j\}$. This is bounded with compact smooth boundary, satisfies ${\bar{V}}_j\subset U$ (indeed $\delta \ge t_j-{\epsilon }_j>{\epsilon }_j$ on its closure), and contains $\{\delta >4{\epsilon }_j\}$. Every compact subset of $U$ is contained in $V_j$ for all sufficiently large $j$. Discard an initial empty $V_j$ if necessary. Nestedness is not required anywhere below. If there are several components, use the same argument componentwise; a compact smooth boundary has a finite graph cover and the proof supplies the same finite-cover regularity on their union.

Choose $s_j<{\epsilon }_j\mathrm{/}4$ tending to zero and mollify the original coefficients and forcing at scale $s_j$ on a neighborhood of ${\bar{V}}_j$: $A_j={\rho }_{s_j}\ast A,{\beta }_j={\rho }_{s_j}\ast \beta ,c_j={\rho }_{s_j}\ast c,f_j={\rho }_{s_j}\ast f\mathrm{.}$ Only values inside $U$ are used there. Consequently $A_j=A_j^T$, $\lambda I\le A_j\le \mathrm{\Lambda }I$, $c_j\le 0$, and the sup bounds of ${\beta }_j,c_j,f_j,D{A}_j$ do not exceed the original corresponding bounds. On every fixed interior ball, their Hölder norms are uniformly bounded and they converge uniformly to the original coefficients/data; $D{A}_j\to DA$ uniformly locally as well. The Hölder bound follows directly by integrating $\mathrm{\mid }F(x-z)-F(y-z)\mathrm{\mid }\le [F{]}_{\alpha }\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }$. Each coefficient is smooth near ${\bar{V}}_j$; its higher derivatives can grow with $j$, which does not matter.

Extend $g$ to a bounded continuous $G$ on ${\mathbb{R}}^n$ using Tietze and set $G_j={\rho }_{s_j}\ast G$. Uniform convergence $G_j\to G$ holds on any fixed compact neighborhood of $\bar{U}$, by uniform continuity there. Also $\mathrm{\parallel }{G}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }G{\mathrm{\parallel }}_{\mathrm{\infty }}$. The lemma gives a smooth classical $u_j$ on ${\bar{V}}_j$ satisfying $L_j{u}_j=f_j\text{in }{V}_j,u_j=G_j\text{on }\mathrm{\partial }{V}_j\mathrm{.}$

Take a slab $a\le x_1\le b$ containing $\bar{U}$, put $B={\sup }_j\mathrm{\parallel }{\beta }_{j,1}{\mathrm{\parallel }}_{\mathrm{\infty }}$, $C_0={\sup }_j\mathrm{\parallel }{c}_j{\mathrm{\parallel }}_{\mathrm{\infty }}$, and choose $\kappa$ with $\lambda {\kappa }^2-B\kappa -C_0\ge 1$. For $\psi =e^{\kappa x_1}$, $L_j\psi \ge e^{\kappa a}=:\eta >0$. Fix $M>e^{\kappa b}$ and put $\mathrm{\Phi }=M-\psi >0$. Then $L_j\mathrm{\Phi }\le -\eta$ for every $j$. The comparison implication used below follows directly: if $z\in C^2(V_j)\cap C({\bar{V}}_j)$, $L_{j}z\ge 0$ and $z\le 0$ on $\mathrm{\partial }{V}_j$, a value $z(x_0)>0$ would, for $ϵ>0$ sufficiently small that $ϵe^{\kappa b}<z(x_0)$, force $z+ϵ\psi$ to have a positive interior maximum. There its gradient is zero and Hessian negative semidefinite, so ellipticity and $c_j\le 0$ give $L_j(z+ϵ\psi )\le 0$. This contradicts $L_j(z+ϵ\psi )\ge ϵ\eta >0$, proving $z\le 0$.

Let $Q=\mathrm{\parallel }G{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $R=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$. On $\mathrm{\partial }{V}_j$, both $u_j-Q-(R\mathrm{/}\eta )\mathrm{\Phi }$ and $-u_j-Q-(R\mathrm{/}\eta )\mathrm{\Phi }$ are nonpositive. Their $L_j$ images are nonnegative, since $c_j\le 0$, $\mathrm{\mid }{f}_j\mathrm{\mid }\le R$ and $L_j\mathrm{\Phi }\le -\eta$. Comparison gives $\mathrm{\mid }{u}_j\mathrm{\mid }\le Q+(R\mathrm{/}\eta )\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{\mathrm{\infty }}=:M_0$ on $V_j$. This constant depends only on the original domain/slab, uniform ellipticity and expanded coefficient sup bounds, $R,Q$. It is independent of $j$, of the smoothness of $\mathrm{\partial }{V}_j$, and of all higher coefficient derivatives.

For every ball $B_{2r}(x)⋐U$, it is contained in $V_j$ for large $j$. Apply the actual interior Schauder theorem to $u_j$ there. All coefficient $C^{\alpha }$ norms and ellipticity constants are uniformly controlled on that fixed ball, $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le M_0$, and the $C^{\alpha }$ norm of $f_j$ is uniformly bounded. Hence $u_j$ is uniformly bounded in $C^{2,\alpha }(B_r(x))$.

Cover $U$ by a countable family of such inner balls with rational centers/radii. Arzelà–Ascoli on each smaller closed ball and diagonal extraction give one subsequence for which $(u_j,D{u}_j,D^2{u}_j)$ converges uniformly on every compact subset of $U$. The fundamental theorem of calculus on coordinate segments identifies the limits with $(u,Du,D^{2}u)$; their compatibility on overlapping balls follows from uniform convergence. Passing each Hessian Hölder difference quotient to the limit proves $u\in C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\alpha }(U)$. Uniform local convergence of all coefficients/data and derivatives gives $Lu=f$ pointwise. The uniform bound gives $\mathrm{\mid }u\mathrm{\mid }\le M_0$.

Fix $q\in \mathrm{\partial }U$. Let $B_R(z)$ be an exterior ball tangent at $q$. Replace it by the smaller ball with center $z^{\mathrm{\prime }}=(z+q)\mathrm{/}2$ and radius $r_0=R\mathrm{/}2$. Its closed ball is contained in $B_R(z)\cup \{q\}$: for $x\mathrm{\ne }q$ on the smaller closed ball, strict triangle inequality gives $\mathrm{\mid }x-z\mathrm{\mid }<R$. Thus $\mathrm{\mid }x-z^{\mathrm{\prime }}\mathrm{\mid }\ge r_0(x\in \bar{U}),\mathrm{\mid }x-z^{\mathrm{\prime }}\mathrm{\mid }=r_0⟺x=q\mathrm{.}$ This strict contact property is necessary; the original exterior ball could touch the boundary at several points.

Write $r=\mathrm{\mid }x-z^{\mathrm{\prime }}\mathrm{\mid }$, $e=(x-z^{\mathrm{\prime }})\mathrm{/}r$, and let $R_1={\max }_{\bar{U}}r$. Choose an integer $m>0$ so large that $(m+2)\lambda -n\mathrm{\Lambda }-R_1{\sup }_j\mathrm{\parallel }{\beta }_j{\mathrm{\parallel }}_{\mathrm{\infty }}\ge 1.$ Define $W_0=r_0^{-m}-r^{-m}$. It is smooth near $\bar{U}$, nonnegative there, vanishes exactly at $q$, and a direct differentiation gives $L_j{W}_0=m{r}^{-m-2}(\mathrm{tr}{A}_j-(m+2)e^T{A}_{j}e+r{\beta }_j\cdot e)+c_j{W}_0\le -m{r}^{-m-2}\le -m{R}_1^{-m-2}\mathrm{.}$ After multiplying by $R_1^{m+2}\mathrm{/}m$, obtain a single $W_q\ge 0$ with $W_q(q)=0$, $W_q>0$ on $\bar{U}\setminus \{q\}$, and $L_j{W}_q\le -1$ for every $j$.

Fix $0<ϵ<1$. Continuity of $G$ at $q$ gives a neighborhood $N$ of $q$ where $\mathrm{\mid }G(x)-g(q)\mathrm{\mid }<ϵ\mathrm{/}2$. Uniform approximation gives $\mathrm{\mid }{G}_j-G\mathrm{\mid }<ϵ\mathrm{/}2$ on $\bar{U}$ for all sufficiently large $j$. Thus $\mathrm{\mid }{G}_j-g(q)\mathrm{\mid }<ϵ$ on $N\cap \bar{U}$. On the compact complementary set $\bar{U}\setminus N$, $W_q$ has a strictly positive minimum; choose $K$ so large that $\mathrm{\mid }{G}_j-g(q)\mathrm{\mid }\le ϵ+K{W}_q\text{on all of }\bar{U}$ for all sufficiently large $j$, and also $K\ge R+C_0(\mathrm{\mid }g(q)\mathrm{\mid }+ϵ)\mathrm{.}$ Apply comparison on $V_j$ to $z_j^{+}=u_j-g(q)-ϵ-K{W}_q,z_j^{-}=-u_j+g(q)-ϵ-K{W}_q\mathrm{.}$ They are nonpositive on $\mathrm{\partial }{V}_j$, and $L_j{z}_j^{+}=f_j-c_j(g(q)+ϵ)-K{L}_j{W}_q\ge -R-C_0(\mathrm{\mid }g(q)\mathrm{\mid }+ϵ)+K\ge 0,$ $L_j{z}_j^{-}=-f_j+c_j(g(q)-ϵ)-K{L}_j{W}_q\ge -R-C_0(\mathrm{\mid }g(q)\mathrm{\mid }+ϵ)+K\ge 0.$ Therefore $\mathrm{\mid }{u}_j(x)-g(q)\mathrm{\mid }\le ϵ+K{W}_q(x)(x\in V_j)$ for all sufficiently large $j$. Passing to the interior limit gives the same bound for $u$ on $U$. Let $x\to q$ from inside $U$ and use $W_q(x)\to 0$; then ${\mathrm{lim\hspace{0.17em}sup}}_{x\to q}\mathrm{\mid }u(x)-g(q)\mathrm{\mid }\le ϵ$. Since $ϵ$ was arbitrary, $u(x)\to g(q)$. Set $u=g$ on $\mathrm{\partial }U$. The resulting function is continuous at every boundary point and in the interior, hence belongs to $C(\bar{U})$.

This proves existence. The exponential-barrier comparison from the residual-stability lemma applied to the difference of two solutions proves uniqueness. Its stated residual estimate applied to $u-h$ proves the stability branch without any regularity of $h$ stronger than $C^2(U)\cap C(\bar{U})$ plus bounded residual. No boundary-correction existence needs to be assumed.

Now set $g=0$ and choose $G=G_j=0$ exactly. Let $u_j$ be the corresponding approximants above. For them, the positive equation expands equivalently to $\mathrm{div}(A_{j}D{u}_j)+q_j\cdot D{u}_j+c_j{u}_j=f_j,q_j={\beta }_j-\mathrm{div}{A}_j\mathrm{.}$ The bounds on $D{A}_j,{\beta }_j$ give ${\sup }_j\mathrm{\parallel }{q}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le Q_1<\mathrm{\infty }$. This is where the original $C^{1,\alpha }$ regularity of $A$ is used; existence in the purely nondivergence formulation required only $C^{\alpha }$ principal coefficients after an appropriate different approximation argument.

Multiply by $u_j$, integrate over $V_j$, and use $u_j=0$ on its smooth boundary: $\lambda \mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }{f}_j{\mathrm{\parallel }}_2\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_2+Q_1\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_2\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_2+\mathrm{\parallel }{c}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_2^2\mathrm{.}$ Young's inequality gives $\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_2^2\le C(\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_2^2)\mathrm{.}$ The uniform supremum estimate with $Q=0$ yields $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_2\le \mathrm{\mid }U{\mathrm{\mid }}^{1\mathrm{/}2}C\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, and $\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_2\le \mathrm{\mid }U{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus ${\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{H^1(V_j)}\le C_U\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ Because $u_j$ is smooth to its smooth boundary and zero there, it lies in $H_0^1(V_j)$. Extend it by zero to $U$; approximation of each class by $C_c^{\mathrm{\infty }}(V_j)\subset C_c^{\mathrm{\infty }}(U)$ proves that this extension belongs to $H_0^1(U)$ and has the same $H^1$ norm.

For completeness, no unproved compactness premise is hidden here. In the Hilbert space $H_0^1(U)$, take the closed span of the countable sequence of extensions. Apply Gram–Schmidt to its listed vectors, discarding zeros, to obtain a finite or countable orthonormal basis of that span. Boundedness, diagonal extraction of the scalar coordinates, and Bessel's inequality give square-summable limiting coordinates and a vector in that closed span. Approximate each test vector by its finite basis sums; the common norm bound then proves convergence against every inner product test vector in that span. For any ambient Hilbert test vector $z$, finite Bessel bounds make the series of its coordinates in this basis converge in the closed span, and its residual is orthogonal to every basis vector and hence to the span; replacing $z$ by this series thus gives the same pairing; Riesz representation therefore proves weak convergence in the ambient Hilbert space, including convergence against the bounded $L^2$ pairings of functions and gradients. Its $L^2$ restriction to every compact interior ball agrees with $u$, since the chosen local-classical subsequence converges uniformly there. Thus the weak limit is the function $u$ almost everywhere on $U$. It follows that $u\in H_0^1(U),\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1(U)}\le C_U\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ Equivalently the weak limit in $H^1(U)$ remains in the closed linear subspace $H_0^1(U)$; the direct closed-span construction already takes place in that subspace.

Passing the original weak equations to the limit against any $\phi \in C_c^{\mathrm{\infty }}(U)$ is legitimate by local uniform convergence of coefficients/data and weak convergence of gradients. The divergence representation above shows that $u$ solves the variational equation; density extends it to all $H_0^1$ tests because $A,q,c,f$ are bounded. This establishes both the finite-energy membership and the correct weak interpretation, rather than inferring them from continuity and zero boundary values alone.

For a general $C^{2,\alpha }$ lifting $G$, apply this zero-boundary result to $v=u-G$ with forcing $f-LG\in C^{0,\alpha }(U)$. Uniqueness of the classical Dirichlet problem identifies that zero-boundary solution with $u-G$.

Suppose $n\ge 2$, $U$ is $C^{2,\alpha }$ and the datum has the stated ambient lifting $G$. Use the smooth interior approximating domains $V_j$ from [F8], which have uniform $C^{2,\alpha }$ graph charts and eventually contain every compact subset of $U$. Write $d_j=\mathrm{dist}({\bar{V}}_j,{\mathbb{R}}^n\setminus U)>0$. Repeat the already proved smooth-coefficient construction at scales $0<r_j<d_j\mathrm{/}4$ tending to zero, so all coefficient, forcing and lifting convolution evaluations near ${\bar{V}}_j$ remain inside $U$. Uniform ellipticity, symmetry, the expanded-potential sign and all global $C^{\alpha }$ bounds are preserved. For the undifferentiated $a,b$ terms, their finite first-derivative bounds yield a global Lipschitz bound on this fixed smooth domain: in each graph patch, sufficiently close points are connected inside a slightly larger patch by a path of length at most $C\mathrm{\mid }x-y\mathrm{\mid }$; a finite cover handles close pairs and the supremum bound handles separated pairs. The prescribed highest-derivative Hölder norms then give the uniform expanded coefficient norms. The lifting mollifications have uniformly bounded $C^{2,\alpha }$ norms and converge uniformly near the closure.

Let $u_j$ be the smooth Dirichlet solutions on $V_j$ with forcing $f_j$ and boundary lifting $G_j$. Their existence and smoothness up to the smooth boundary were proved by the Sobolev-bootstrap helper before using any boundary Schauder estimate. The exponential comparison gives a uniform supremum bound. Apply [F8] to the already smooth zero-boundary functions $w_j=u_j-G_j$, whose forcings $f_j-L_j{G}_j$ have uniformly bounded $C^{\alpha }$ norms. The domain geometry and coefficient controls are uniform, so ${\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{C^{2,\alpha }({\bar{V}}_j)}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{0,\alpha ;U}+\mathrm{\parallel }G{\mathrm{\parallel }}_{2,\alpha ;\bar{U}})$.

The existing interior compactness and exterior-ball boundary comparison arguments give a subsequence converging locally in $C^2(U)$ to the unique classical Dirichlet solution $u$. The displayed global derivative bounds pass to this local limit. For any fixed distinct $x,y\in U$, both points eventually lie in $V_j$, so their second-derivative Hölder quotient passes to the limit with the same common bound. Hence $D^{2}u$ is bounded and globally $\alpha$-Hölder on $U$, and extends continuously to the closure. The function and gradient extend continuously as well: bounded first and second derivatives give Lipschitz bounds along the controlled-length local interior paths in each graph patch, and separated pairs are bounded by the global suprema. These extensions are the continuous extensions of the interior derivatives. Consequently $u\in C^{2,\alpha }(\bar{U})$ with the stated data estimate, obtained entirely from smooth approximants without assuming the original solution's boundary Hölder norm. Together with the parent's one-dimensional argument and the prior local existence, uniqueness, stability and energy steps, all branches follow.