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On a fixed smooth bounded domain, zero-boundary solutions with forcing in a bounded subset of $C^{0,\mu }$ are uniformly bounded in $C^{2,\mu }$ by a maximum-principle barrier and the global Schauder estimate. The Arzelà-Ascoli theorem gives convergence in $C^2$. A closed Hölder-norm constraint passes to the limit by lower semicontinuity of a Hölder seminorm, so the constrained solution set is compact rather than merely relatively compact.

For a fixed uniformly elliptic operator with $C^{0,\mu }$ coefficients on a bounded smooth connected domain, nonnegative solutions with zero Dirichlet boundary condition satisfy the displayed estimate when ${\Omega }^{\prime }$ is nonempty and compactly contained in $\Omega$. If it failed, normalize the global supremum to one. The forcing and the interior infimum then tend to zero. The global Schauder estimate and compact embedding of Hölder spaces produce a nonzero homogeneous limit with an interior zero, contradicting the strong minimum principle for elliptic operators. An arbitrary bounded zeroth-order coefficient is allowed: replace $c$ by $\min (c,0)$ for the positivity argument. Keeping global boundary control prevents the normalized maximum from disappearing in the limit.

For smooth non-increasing $V$ and an ordered subsolution and supersolution, begin with $u_0={\phi }^{-}$ and solve successive Poisson equations with forcing $V(u_k)$ and the fixed boundary data. The weak maximum principle for elliptic operators applied to the negatives of consecutive differences givesThe global Schauder estimate and the Hölder interpolation inequality bound the iterates uniformly in $C^{2,\alpha }$. Their bounded monotone sequence converges, and compact embedding of Hölder spaces permits passing to the equation. The uniform Hölder seminorm bounds retain $C^{2,\alpha }$ regularity of the limit.

Use the usual fractional Dirichlet domain scale: $A=-{\Delta }_D$ on $H=L^2(\Omega )$, and $H_{\rho }=D(A^{\rho })$. Its Sobolev embedding theorem givesSuch a positive $\beta$ exists precisely because the assumed $\delta$ is greater than $n/4$. Consequently $v=u_t$ is continuous in time with values in this Hölder space on every compact positive-time interval. This embedding alone gives no two spatial derivatives for $v$; those must come from the equation.

First establish a bound that will also justify the nonlinear regularity steps. Put $F(s)=\lambda s-s^3$ andThe constants $K$ and $-K$ enclose both the initial data and the zero Dirichlet boundary data. Moreover $F(K)\le 0$ and $F(-K)\ge 0$, so they are a supersolution and a subsolution for the parabolic comparison principle. A direct energy proof avoids assuming pointwise continuity at time zero. Let $h=(u-K{)}_{+}$. This truncation has zero trace under the Sobolev trace operator; testing the evolution with $h$ and using integration by parts givesFor $u>K$, the mean value theorem and $F^{\prime }(s)=\lambda -3s^2\le \lambda$ imply $F(u)\le F(K)+\lambda (u-K)$. ThereforeThe Gronwall inequality gives $h=0$, since its initial L2 norm vanishes. For the initially rough datum, apply the inequality from $\epsilon >0$ and let $\epsilon \downarrow 0$, using $u(\epsilon )\to u_0$ in L2 space. The same argument for $-u$, which satisfies the same equation because $F$ is odd, provesThis is the invariant interval for a cubic reaction-diffusion equation; the energy calculation is valid by the Sobolev chain rule and approximation even before the stronger pointwise regularity is established.

Now fix $0<\tau <t_0$ and, if necessary, reduce the positive Hölder exponent $\beta$. Rearranging the evolution gives the elliptic identityIts right side is bounded on $[\tau ,t_0]\times \Omega$, because $v\in C([\tau ,t_0],C^{0,\beta })$ and $|u|\le K$. Boundary elliptic regularity first gives $u(t)\in W^{2,p}(\Omega )$ for every finite $p$, uniformly on this time interval. For $p>n$, the Sobolev embedding theorem makes $u(t)$ spatially continuously differentiable, with a positive Hölder exponent. Thus $F(u(t))$ lies in $C^{0,\beta }$ after reducing $\beta$ if necessary. The global Schauder estimate then gives $u(t)\in C^{2,\beta }(\overline{\Omega })$.

These are continuous-in-time, not just separate-time, regularity statements. Since $v\in C(C^{0,\beta })$, the identity $u(t)-u(s)={\int }_s^{t}v(r)dr$, initially an identity in L2 space, holds in $C^{0,\beta }$ once $u(s)$ is known to belong to that space. Hence $u$ is locally Lipschitz in time into $C^{0,\beta }$. Apply the global Schauder estimate to the difference of the two elliptic identities. Continuity of $v$ and of $F(u)$ into $C^{0,\beta }$ proves $u\in C([\tau ,t_0],C^{2,\beta })$.

To gain the spatial regularity of the time derivative itself, take time difference quotients of the evolution. If $d_h(t)=[u(t+h)-u(t)]/h$, their equations arewhereThe coefficients have uniformly bounded spatial Hölder norms and temporal Hölder norms on compact positive-time cylinders: this follows from the spatial regularity just proved and the local time Lipschitz bound in $C^{0,\beta }$. Also $d_h={\int }_0^{1}v(t+sh)ds$ is uniformly bounded and tends to $v$ in $C^{0,\beta }$. Positive-time parabolic Schauder estimates, applied away from the lower time edge, give uniform $C^{1+\beta /2,2+\beta }$ bounds for $d_h$, with a slightly smaller exponent if needed. Passing to the limit proves that $v$ is continuous into $C^2(\overline{\Omega })$ and satisfiesThe Dirichlet boundary condition for $v$ follows also by differentiating the identically zero boundary value in $C^0$. Since $\tau$ was arbitrary, this proves the requested, genuinely stronger regularity:This is positive-time smoothing for a semilinear heat equation; it is the extra parabolic step for $v$, not just the Sobolev embedding theorem, that gives the $C^1$ topology asserted here.

Finally, a uniform bound excludes a finite maximal existence time for this semilinear partial differential equation. Restart at any positive time $s$, using the heat semigroup on the Banach space $X=\{g\in C(\overline{\Omega }):g{|}_{\partial \Omega }=0\}$ with the supremum norm. Its variation-of-constants formula isThe heat semigroup is a contraction semigroup on $X$. On the ball of radius $K+1$, $F$ is Lipschitz with constant at most $|\lambda |+3(K+1{)}^2$, and bounded by $|\lambda |(K+1)+(K+1{)}^3$. Choose a time length $\eta >0$ so that the integral is at most $1$ and its Lipschitz constant is less than $1$. The Banach fixed-point theorem then gives a local solution on $[s,s+\eta ]$, with $\eta$ independent of $s$, because $\Vert u(s){\Vert }_{\infty }\le K$. If a finite maximal time existed, choosing $s$ less than $\eta$ before it would extend the same solution past it, a contradiction. Uniqueness identifies this continuation with the original solution, and positive-time smoothing restores the classical solution on each extended interval. The solution is globally defined and remains between $-K$ and $K$.

We prove compactness of zero-boundary Poisson solutions. Here the printed $\partial \Omega$ must mean $\partial B_1$; otherwise the boundary condition refers to an unspecified set. Use this intended boundary condition throughout. Let $f=\Delta u$. A maximum-principle barrier gives a uniform bound on $u$: with $\psi (x)=(1-|x{|}^2)/(2n)$, we have $\Delta \psi =-1$ and $\psi =0$ on $\partial B_1$. Since $|f|\le 1$,The weak maximum principle gives $-\psi \le u\le \psi$, so $\Vert u{\Vert }_{\infty }\le 1/(2n)$.

Standard elliptic regularity on the smooth ball first gives $u\in C^{2,\mu }(\overline{B_1})$. The global Schauder estimate then givesThus the functions and all their derivatives through order two are uniformly bounded and equicontinuous on the closed ball. The Arzelà-Ascoli theorem gives a subsequence converging uniformly along with all these derivatives. Integration along line segments identifies the limiting derivatives, so the convergence is in $C^2(\overline{B_1})$.

It remains to keep the limit in the set, rather than merely proving relative compactness. If $u_j\to u$ in $C^2$, then the zero boundary values pass to the limit and $f_j=\Delta u_j\to f=\Delta u$ uniformly. For each pair of distinct points,Taking suprema gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$, while $\Vert f_j{\Vert }_{\infty }\to \Vert f{\Vert }_{\infty }$. Hence $\Vert f{\Vert }_{C^{0,\mu }}\le 1$. The limit belongs to the set, proving sequential compactness in $C^2(\overline{B_1})$. The fixed positive Hölder exponent supplies the compactness; a bound on the supremum norm of $\Delta u$ alone would not supply this argument.

Take ${\Omega }^{\prime }$ nonempty and compactly contained in $\Omega$. The argument is a compactness proof of a nonnegative elliptic solution estimate, with the global Schauder estimate retaining control all the way to the boundary. The operator and the Hölder space exponent are fixed throughout.

Suppose the estimate fails. For each integer $k$ there is an admissible nonnegative solution $u_k$, with forcing $f_k$, such thatPut $v_k=u_k/M_k$ and $g_k=f_k/M_k$. Then $0\le v_k\le 1$, ${\max }_{\overline{\Omega }}{v}_k=1$, $v_k=0$ on the boundary, $L{v}_k=g_k$, andGlobal elliptic regularity and the global Schauder estimate for zero Dirichlet boundary condition giveThis estimate includes the $C^0$ term, so it does not require invertibility of $L$ or a sign restriction on $c$. The compact embedding of Hölder spaces, or the Arzelà-Ascoli theorem applied through second partial derivatives, gives a subsequence converging in $C^2(\overline{\Omega })$ to $v$. Thus $v\ge 0$, $Lv=0$, $v=0$ on the boundary, and $\max v=1$.

Choose $x_k\in {\Omega }^{\prime }$ with $v_k(x_k)\le {\inf }_{{\Omega }^{\prime }}{v}_k+1/k$. A further subsequence has $x_k\to x_{\ast }\in \overline{{\Omega }^{\prime }}\subset \Omega$. Uniform convergence yields $v(x_{\ast })=0$. Part (b) says that a nonnegative homogeneous solution on the connected domain is either zero everywhere or strictly positive inside. Both alternatives contradict, respectively, $\max v=1$ and $v(x_{\ast })=0$. ThereforeThe interior infimum and the forcing control the global supremum. Global boundary control is crucial here: an interior-only compactness argument could lose the normalized maximum at the boundary.

The global Schauder estimate on a bounded $C^{2,\alpha }$ domain isHere $C$ depends on $n,\alpha$, the domain's boundary regularity and $\Vert c{\Vert }_{C^{0,\alpha }}$, but not on the particular solution or data. The $C^0$ term is needed before uniqueness has been established. The given $\psi$ is an extension of the boundary data to the closure.

For $c\le 0$, choose $d$ with $\Omega \subset \{|x_1|<d\}$ and set $M={\sup }_{\partial \Omega }|\psi |$, $F={\sup }_{\Omega }|f|$. The hinted functionis nonnegative, dominates $|\psi |$ on the boundary, and satisfiessince $x_1+d>0$. Therefore $(\Delta +c)(u-b)=f-(\Delta +c)b\ge 0$ and $(\Delta +c)(-u-b)=-f-(\Delta +c)b\ge 0$. Both functions are nonpositive on the boundary. The weak maximum principle for elliptic operators gives $|u|\le b$, hence the supremum norm barrier for an elliptic Dirichlet problem:Unlike the full Schauder estimate, this last constant depends only on a slab containing $\Omega$, and is independent of how negative $c$ is.

For zero Dirichlet boundary condition, the global Schauder estimate isThe constant depends on $n,\alpha$, domain regularity, the ellipticity lower bound, and coefficient $C^{0,\alpha }$ norms. No sign restriction on $c$ is needed. Nonzero boundary data contribute the $C^{2,\alpha }$ norm of an extension.

Take ${\Omega }^{\prime }$ nonempty. If the bound failed, normalization would give nonnegative $v_j$ with supremum one, zero boundary values, ${\inf }_{{\Omega }^{\prime }}{v}_j\to 0$, and $\Vert L{v}_j{\Vert }_{C^{0,\alpha }}\to 0$. The global Schauder estimate bounds them in the Hölder space $C^{2,\alpha }$. By compact embedding of Hölder spaces, a subsequence converges in $C^{2,\beta }$, $0<\beta <\alpha$, to $v\ge 0$ with $Lv=0$, zero boundary data and supremum one.

Points approaching the infima have a subsequence converging in $\overline{{\Omega }^{\prime }}\subset \Omega$. Uniform convergence gives an interior zero of $v$, and part (c), the strong minimum principle for elliptic operators after replacing $c$ by $\min (c,0)$, gives $v=0$. This contradicts its supremum one, proving

Failure of the bound would give normalized $v_j$ with $\Vert v_j{\Vert }_{\infty }=1$, zero Dirichlet boundary condition, and $\Vert L{v}_j{\Vert }_{C^{0,\alpha }}\to 0$. The global Schauder estimate bounds them in $C^{2,\alpha }$. By compact embedding of Hölder spaces, a subsequence converges in $C^{2,\beta }$ to $v$ with $Lv=0$, zero boundary data and $\Vert v{\Vert }_{\infty }=1$. The uniform Hölder seminorm bounds pass to the limit, so $v\in C^{2,\alpha }$. This contradicts the trivial null space, provingThe global Schauder estimate then gives $\Vert u{\Vert }_{C^{2,\alpha }}\le K\Vert f{\Vert }_{C^{0,\alpha }}$, the bounded-inverse estimate used next.

For a sequence trapped between the fixed barriers, let $M$ bound their absolute values and put $A={\sup }_{|s|\le M}|V^{\prime }(s)|$. The mean value theorem givesThe global Schauder estimate, including the fixed boundary data $\psi$, and $\Vert u_k{\Vert }_{\infty }\le M$ implyApply the supplied Hölder interpolation inequality with $KA\epsilon \le 1/2$. Its remaining supremum-norm term is bounded by $M$, yieldingIf $A=0$, the stronger bound without a previous-iterate term holds directly. The constant depends on the fixed domain, exponent, barriers, boundary data and $V$, independently of $k$.

The stated assumptions alone do not imply regularity up to the boundary: continuous boundary data need not have two Hölder derivatives. A standard sufficient set of hypotheses for the global Schauder estimate iswith $a^{ij},b^i,c\in C^{0,\alpha }(\overline{\Omega })$ and uniform ellipticity. Then the global Schauder estimate givesand bounds its norm by the forcing, boundary data, and $C^0$ norm.

For $w\in X_{\delta }$, hypothesis (2) gives $F(w)\in C^{0,\alpha }(\overline{\Omega })$. Thus $f-F(w)$ is Hölder continuous and the classical Dirichlet problem for the Poisson equation has a unique solution $u=T(w)\in C^{2,\alpha }(\overline{\Omega })$. The global Schauder estimate givesThe maximum estimate for the Poisson problem also givesThese estimates prove that $T$ is well defined and give the requested norm control.

The global Schauder estimate isThe constant depends on the dimension, $\alpha$, the domain and its boundary regularity, the ellipticity constants, and the $C^{0,\alpha }$ norms of the coefficients, but not on $u$, $f$, or $\phi$.

Let $v$ range over a bounded subset of $C^{1,\beta }$. All jets $(x,v,Dv)$ then lie in one compact set, so the composed coefficients have uniform $C^{0,\alpha \beta }$ bounds and one ellipticity constant. The weak maximum principle for elliptic operators bounds $T(v)$ in $C^0$ by the boundary data and the bounded forcing term. The global Schauder estimate consequently bounds $T(v)$ in $C^{2,\alpha \beta }$.

The compact embedding of Hölder spacesthen makes the image relatively compact. Hence $T$ is a compact operator on $C^{1,\beta }(\overline{\Omega })$.

The Leray-Schauder fixed point theorem says that a continuous compact map $T$ on a Banach space has a fixed point if the homotopy setis bounded. If $u=tT(u)$ with $t>0$, multiplying the equation for $T(u)$ by $t$ shows that $u$ solvesThe case $t=0$ gives $u=0$. Therefore it is enough to prove one uniform $C^{1,\beta }$ estimate for all solutions of this family and all $t\in [0,1]$. The theorem then yields a fixed point $u=T(u)$, which solves the original quasilinear problem.

Initially the construction gives $u\in C^{2,\alpha \beta }$. This makes $u$ and $Du$ Lipschitz, so composing the original $C^{0,\alpha }$ coefficients with the jet of $u$ produces $C^{0,\alpha }$ coefficients. A second application of the global Schauder estimate gives $u\in C^{2,\alpha }(\overline{\Omega })$. The fixed-point theorem is an existence result and supplies no uniqueness.