Proof of Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem

Conventions. From the setting we use the real numbers with their order, Euclidean space with its distance $d_{E}$ and notion of openness, and the notions of semicontinuity and local extrema. Elementary order facts are those of Elementary Order Arithmetic in an Ordered Field; the non-strict compatibility of $\le$ with addition is an axiom of the ordered field $\mathbb{R}$, whose order is a total order, so that its reflexivity, antisymmetry, transitivity and totality are axioms of that definition. If $a<b$ fails then $b\le a$: by totality $a\le b$ or $b\le a$, and in the first case $a\ne b$ would give $a<b$, so $a=b$ and $b\le a$ by reflexivity.

Throughout, $\Omega\subseteq\overline{\Omega}$ and $\partial\Omega=\overline{\Omega}\setminus\Omega$, as recorded in Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem; in particular $\Omega$ and $\partial\Omega$ are disjoint. For $\underline{w}\in\mathcal{G}$ the restriction $\underline{w}|_{\Omega}$ is a viscosity subsolution of $F$ on $\Omega$, by the definition of a subsolution up to the boundary.

Step 1: $W$ and its envelopes. Every $x\in\overline{\Omega}$ satisfies $\underline{u}(x)\le W(x)\le\overline{u}(x)$: the first because $\underline{u}\in\mathcal{G}$, the second because $\overline{u}(x)$ is an upper bound of the set whose least upper bound is $W(x)$. Hence $-M\le W(x)\le M$ for every $x\in\overline{\Omega}$, by claim 6 of Properties of the Absolute Value in an Ordered Field applied to the hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (ii) and transitivity. Taking any positive radius, this shows that $W$ is bounded above and below near each point of $\overline{\Omega}$, so $W^{*}$ and $W_{*}$ are defined.

Step 2: the envelopes agree with the boundary data on $\partial\Omega$. The functions $\underline{u}$ and $\overline{u}$ are likewise bounded above and below near each point of $\overline{\Omega}$, so all four envelopes occurring below are defined. From $W(x)\le\overline{u}(x)$ for every $x$ and claim 6 of Properties of the Upper Semicontinuous Envelope we get $W^{*}(x)\le\overline{u}^{*}(x)$, and from $\underline{u}(x)\le W(x)$ and claim 7 of Properties of the Lower Semicontinuous Envelope, by Duality we get $\underline{u}_{*}(x)\le W_{*}(x)$, for every $x\in\overline{\Omega}$. Let $x\in\partial\Omega$. By hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (ii),

$$g(x)=\underline{u}_{*}(x)\le W_{*}(x)\le W(x)\le W^{*}(x)\le\overline{u}^{*}(x)=g(x),$$

using claim 2 of Properties of the Lower Semicontinuous Envelope, by Duality and claim 1 of Properties of the Upper Semicontinuous Envelope for the two middle inequalities. By transitivity and antisymmetry,

$$(1)\qquad W_{*}(x)=W(x)=W^{*}(x)=g(x)\qquad\text{for every }x\in\partial\Omega .$$

Step 3: the envelopes are computed inside $\Omega$ from the restriction. Let $x\in\Omega$. Since $\Omega$ is open, claim 4 of The Interior is the Largest Open Subset and claim 1 of Interior Points in the Metric Topology are Exactly the Centres of Contained Closed Balls give a positive $r$ with $\{y\in\mathbb{R}^{n}:d_{E}(y,x)\le r\}\subseteq\Omega$; hence the set $S_{r}=\{y\in\overline{\Omega}:d_{E}(y,x)\le r\}$ coincides with $\{y\in\Omega:d_{E}(y,x)\le r\}$ and $W|_{S_{r}}=(W|_{\Omega})|_{S_{r}}$. Applying claim 4 of Semicontinuity and the Semicontinuous Envelopes are Local Notions twice, once with the ambient set $\overline{\Omega}$ and once with the ambient set $\Omega$, and evaluating at $y=x$, we obtain $W^{*}(x)=(W|_{S_{r}})^{*}(x)=(W|_{\Omega})^{*}(x)$. The same argument with claim 5 of that lemma gives $W_{*}(x)=(W|_{\Omega})_{*}(x)$. Thus

$$(2)\qquad W^{*}(x)=(W|_{\Omega})^{*}(x)\quad\text{and}\quad W_{*}(x)=(W|_{\Omega})_{*}(x)\qquad\text{for every }x\in\Omega .$$

Step 4: $W$ is a viscosity subsolution of $(F,g)$. Let $\mathcal{F}$ be the set of the restrictions $\underline{w}|_{\Omega}$ with $\underline{w}\in\mathcal{G}$, a nonempty set of viscosity subsolutions of $F$ on $\Omega$. It is locally uniformly bounded above in the sense of The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution (i): given $x\in\Omega$, take $c=M$ and any positive radius, since every $\underline{w}\in\mathcal{G}$ satisfies $\underline{w}(y)\le\overline{u}(y)\le M$ at every $y\in\overline{\Omega}$, by the definition of $\mathcal{G}$, hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (ii), claim 6 of Properties of the Absolute Value in an Ordered Field and transitivity. Its pointwise supremum is $W|_{\Omega}$, by the definition of $W$. Hence The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution shows that $(W|_{\Omega})^{*}$ is a viscosity subsolution of $F$ on $\Omega$, and by (2) this function is $W^{*}|_{\Omega}$.

By claim 2 of Properties of the Upper Semicontinuous Envelope the function $W^{*}$ is upper semicontinuous on $\overline{\Omega}$, so $W^{*}$ is a viscosity subsolution of $F$ up to the boundary of $\Omega$; and $W^{*}(x)=g(x)$ for $x\in\partial\Omega$ by (1). Therefore $W^{*}$ is a viscosity subsolution of $(F,g)$. By claim 1 of Properties of the Upper Semicontinuous Envelope and Step 1 we have $\underline{u}(x)\le W(x)\le W^{*}(x)$ for every $x$, and hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (i), applied to the subsolution $W^{*}$ and the supersolution $\overline{u}$, gives $W^{*}(x)\le\overline{u}(x)$. Hence $W^{*}\in\mathcal{G}$, so $W^{*}(x)\le W(x)$ for every $x\in\overline{\Omega}$, $W(x)$ being an upper bound of the values of the members of $\mathcal{G}$ at $x$. With $W\le W^{*}$ and antisymmetry this gives $W=W^{*}$. In particular $W$ is a viscosity subsolution of $(F,g)$.

Step 5: $W_{*}$ restricted to $\Omega$ is a viscosity supersolution of $F$. By claim 3 of Properties of the Lower Semicontinuous Envelope, by Duality the function $W_{*}$ is lower semicontinuous on $\overline{\Omega}$, hence its restriction to $\Omega$ is lower semicontinuous on $\Omega$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. Suppose, seeking a contradiction, that $W_{*}|_{\Omega}$ is not a viscosity supersolution of $F$ on $\Omega$. By the definition of a viscosity supersolution the failure must then be of the test inequality: there are $\hat x\in\Omega$ and a function $\varphi:\Omega\to\mathbb{R}$ of class $C^{2}$ on $\Omega$ such that $W_{*}|_{\Omega}-\varphi$ has a local minimum at $\hat x$ relative to $\Omega$ and $0\le F(\hat x,W_{*}(\hat x),D\varphi(\hat x),D^{2}\varphi(\hat x))$ fails, hence, by the remark in the Conventions,

$$F\bigl(\hat x,W_{*}(\hat x),D\varphi(\hat x),D^{2}\varphi(\hat x)\bigr)<0 .$$

The function $W|_{\Omega}$ is a viscosity subsolution of $F$ on $\Omega$ by Step 4 and (2), being $W^{*}|_{\Omega}$; it is bounded below near each point of $\Omega$ by Step 1, and its lower semicontinuous envelope is $W_{*}|_{\Omega}$ by (2). So The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality applies to it at $\hat x$ with the test function $\varphi$.

Since $\Omega$ is open, choose as in Step 3 a positive $\kappa$ with $\{y\in\mathbb{R}^{n}:d_{E}(y,\hat x)\le\kappa\}\subseteq\Omega$, and let $U_{\kappa}:\Omega\to\mathbb{R}$ be as provided by The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality for this $\kappa$. Define $\widehat{W}:\overline{\Omega}\to\mathbb{R}$ by $\widehat{W}(x)=U_{\kappa}(x)$ for $x\in\Omega$ and $\widehat{W}(x)=W(x)$ for $x\in\partial\Omega$; this is well defined because $\Omega$ and $\partial\Omega$ are disjoint and cover $\overline{\Omega}$.

$\widehat{W}$ agrees with $W$ off the ball of radius $\kappa$. If $x\in\Omega$ and $\kappa\le d_{E}(x,\hat x)$ then $\widehat{W}(x)=U_{\kappa}(x)=W(x)$ by The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality (iii). If $x\in\partial\Omega$ then $x\notin\Omega$, so $d_{E}(x,\hat x)\le\kappa$ fails and hence $\kappa<d_{E}(x,\hat x)$ by the remark in the Conventions, while $\widehat{W}(x)=W(x)$ by definition.

$\widehat{W}$ is upper semicontinuous on $\overline{\Omega}$. Let $x\in\overline{\Omega}$. If $x\in\Omega$, choose as in Step 3 a positive $r$ with $\{y\in\mathbb{R}^{n}:d_{E}(y,x)\le r\}\subseteq\Omega$; then the set $S_{r}=\{y\in\overline{\Omega}:d_{E}(y,x)\le r\}$ is contained in $\Omega$, on which $\widehat{W}$ agrees with $U_{\kappa}$, and $U_{\kappa}$ is upper semicontinuous on $\Omega$, being a viscosity subsolution there by The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality (i); so $\widehat{W}|_{S_{r}}$ is upper semicontinuous at $x$ relative to $S_{r}$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, and claim 1 of Semicontinuity and the Semicontinuous Envelopes are Local Notions gives that $\widehat{W}$ is upper semicontinuous at $x$ relative to $\overline{\Omega}$. If $x\in\partial\Omega$, put $\varepsilon=d_{E}(x,\hat x)-\kappa$, which is positive by the previous paragraph and claim 1 of Elementary Order Arithmetic in an Ordered Field, and let $\rho=\tfrac{\varepsilon}{2}$, positive by claim 8 of that lemma. Every $y\in\overline{\Omega}$ with $d_{E}(y,x)\le\rho$ satisfies, by the triangle inequality of a metric,

$$d_{E}(x,\hat x)\le d_{E}(x,y)+d_{E}(y,\hat x),$$

so $\kappa+\varepsilon-\rho\le d_{E}(y,\hat x)$ and hence $\kappa<d_{E}(y,\hat x)$, whence $\widehat{W}(y)=W(y)$ by the previous paragraph. Since $W=W^{*}$ is upper semicontinuous on $\overline{\Omega}$ by Step 4, claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions and claim 1 of Semicontinuity and the Semicontinuous Envelopes are Local Notions again give that $\widehat{W}$ is upper semicontinuous at $x$ relative to $\overline{\Omega}$.

$\widehat{W}$ belongs to $\mathcal{G}$. Its restriction to $\Omega$ is $U_{\kappa}$, a viscosity subsolution of $F$ on $\Omega$, so $\widehat{W}$ is a viscosity subsolution of $F$ up to the boundary of $\Omega$; and $\widehat{W}(x)=W(x)=g(x)$ for $x\in\partial\Omega$ by (1), so $\widehat{W}$ is a viscosity subsolution of $(F,g)$. By The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality (ii) we have $W(x)\le\widehat{W}(x)$ for $x\in\Omega$, and equality holds for $x\in\partial\Omega$; hence $\underline{u}(x)\le W(x)\le\widehat{W}(x)$ for every $x\in\overline{\Omega}$, and hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (i) applied to $\widehat{W}$ and $\overline{u}$ gives $\widehat{W}(x)\le\overline{u}(x)$. So $\widehat{W}\in\mathcal{G}$ and therefore $\widehat{W}(x)\le W(x)$ for every $x\in\overline{\Omega}$.

But The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality (ii) also provides $x\in\Omega$ with $W(x)<U_{\kappa}(x)=\widehat{W}(x)$, and with $\widehat{W}(x)\le W(x)$ the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field gives $W(x)<W(x)$, contradicting the irreflexivity of the strict order. Hence $W_{*}|_{\Omega}$ is a viscosity supersolution of $F$ on $\Omega$.

Step 6: conclusion. By Step 5 and the lower semicontinuity of $W_{*}$ on $\overline{\Omega}$, the function $W_{*}$ is a viscosity supersolution of $F$ up to the boundary of $\Omega$, and $W_{*}(x)=g(x)$ for $x\in\partial\Omega$ by (1); so $W_{*}$ is a viscosity supersolution of $(F,g)$. Hypothesis Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem (i), applied to the subsolution $W$ of Step 4 and to the supersolution $W_{*}$, gives $W(x)\le W_{*}(x)$ for every $x\in\overline{\Omega}$; and $W_{*}(x)\le W(x)$ by claim 2 of Properties of the Lower Semicontinuous Envelope, by Duality. By antisymmetry $W=W_{*}$, so $W$ is a viscosity supersolution of $(F,g)$ as well. Being both a viscosity subsolution and a viscosity supersolution of $(F,g)$, the function $W$ is a viscosity solution of the Dirichlet problem $(F,g)$. $\blacksquare$