An Inequality for Hölder Continuous Functions Generalizing a Result of Carlo Miranda

Abstract

We prove an inequality for the Hölder continuity of a continuously differentiable function in a bounded open Lipschitz set, which generalizes an inequality of Carlo Miranda in a bounded open strict hypograph of a function of class C ^{1, α} for some α ∈ ]0, 1] and which enables to simplify a proof of a result of Carlo Miranda for layer potentials with moment in a Schauder space.

Appendix

Proof of Lemma 10.1

If x ∈ ∂Ω, then there exists a coordinate cylinder C(x, R _x, r _x, δ _x) around x and a corresponding function $\tilde {\gamma }_x$ which represents ∂Ω in C(x, R _x, r _x, δ _x). Now let

$$\displaystyle \begin{aligned} \delta_x^{\prime}\in]0,\min\{\delta_x,\delta_\ast\}[\,. \end{aligned}$$

Since $\tilde {\gamma }_x$ is continuous, there exists $r_{x}^{\prime }\in ]0,+\infty [$ such that

$$\displaystyle \begin{aligned} r_{x}^{\prime}<\min\{ r_x,r_\ast\}\,, \qquad |\tilde{\gamma}_x(\eta)|<\frac{1}{2}\delta_x^{\prime} \qquad \forall \eta\in {\mathbb{B}}_{n-1}(0,r_{x}^{\prime})\,. \end{aligned}$$

Then $C(x,R_x,r_x^{\prime },\delta _x^{\prime })$ is a coordinate cylinder for Ω around x and the restriction $\tilde {\gamma }_{x|{\mathbb {B}}_{n-1}(0,r_{x}^{\prime })}$ represents ∂Ω in $C(x,R_x,r_x^{\prime },\delta _x^{\prime })$. By definition of C ^{0, α}-norm, we have

$$\displaystyle \begin{aligned} \| \tilde{\gamma}_{x|{\mathbb{B}}_{n-1}(0,r_{x}^{\prime})} \|{}_{C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r_{x}^{\prime})})} \leq \| \tilde{\gamma}_{x|{\mathbb{B}}_{n-1}(0,r_{x})} \|{}_{C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r_{x})})}\,. \end{aligned}$$

Since

$$\displaystyle \begin{aligned} \left\{x+R_x^t\left( {\mathbb{B}}_{n-1}(0,r_{x}^{\prime}/4) \times]-\delta_x^{\prime},\delta_x^{\prime}[ \right) \right\}_{x\in\partial\varOmega} \end{aligned}$$

is an open cover of ∂Ω and ∂Ω is compact there exists a finite family $\{x^{(j)}\}_{j=1}^k$ of points of ∂Ω such that

$$\displaystyle \begin{aligned} \partial\varOmega\subseteq \bigcup_{j=1}^k \left[x^{(j)}+R_{x^{(j)}}^t\left( {\mathbb{B}}_{n-1}(0,r_{x^{(j)}}^{\prime}/4) \times] -\delta_{x^{(j)}} ^{\prime},\delta_{x^{(j)}}^{\prime}[ \right)\right]\,. \end{aligned}$$

To shorten our notation, we find convenient to set

$$\displaystyle \begin{aligned} R_j\equiv R_{x^{(j)}}\,,\qquad r_j^{\prime}\equiv r_{x^{(j)}}^{\prime}\,\qquad \delta_j^{\prime}\equiv\delta_{x^{(j)}}^{\prime}\,,\qquad \gamma_j\equiv \tilde{\gamma}_{ x^{(j)} |{\mathbb{B}}_{n-1}(0,r_{ x^{(j)}}^{\prime})} \end{aligned}$$

for all j ∈{1, …, k}. Next we choose

$$\displaystyle \begin{aligned} \delta\in]0,\frac{1}{4}\min_{j\in\{1,\dots,k\}}\delta_j^{\prime}[\,. \end{aligned}$$

By the uniform continuity of the functions γ _j, there exists

$$\displaystyle \begin{aligned} r\in ]0,\frac{1}{4}\min_{j\in\{1,\dots,k\}}r_j^{\prime}[ \end{aligned}$$

such that

$$\displaystyle \begin{aligned} |\gamma_j(\eta_1)-\gamma_j(\eta_2)|<\delta/2 \qquad {\text{whenever}}\ \eta_1, \eta_2\in {\mathbb{B}}_{n-1}(0,r_j)\,,\ |\eta_1-\eta_2|<r \end{aligned} $$

(10.22)

for all j ∈{1, …, k}. Next we fix an arbitrary x ∈ ∂Ω and we define a coordinate cylinder for Ω around x. Let j ∈{1, …, k} be such that

$$\displaystyle \begin{aligned} x\in x^{(j)}+R_j^t\left( {\mathbb{B}}_{n-1}(0,r_j^{\prime}/4) \times]-\delta_j^{\prime},\delta_j^{\prime}[\right) \,. \end{aligned}$$

Then there exists $\eta _x\in {\mathbb {B}}_{n-1}(0,r_j^{\prime }/4)$ such that

$$\displaystyle \begin{aligned} x=x^{(j)}+R_j^t( \eta_x,\gamma_j(\eta_x))^t\,. \end{aligned}$$

Since $r<r_j^{\prime }/4$, we have

$$\displaystyle \begin{aligned} {\mathbb{B}}_{n-1}(\eta_x,r)\subseteq {\mathbb{B}}_{n-1}(0,r_j^{\prime}/2)\,. \end{aligned}$$

Since $\delta <\delta _j^{\prime }/4$, we have

$$\displaystyle \begin{aligned} ]\gamma_j(\eta_x)-\delta ,\gamma_j(\eta_x)+\delta [ \subseteq ]-(\delta_j^{\prime}/2)-\delta,(\delta_j^{\prime}/2)+\delta[ \subseteq ]-(3\delta_j^{\prime}/4),(3\delta_j^{\prime}/4)[\,. \end{aligned}$$

Next we set

$$\displaystyle \begin{aligned} \gamma_x(\eta)\equiv \gamma_j(\eta_x+\eta)-\gamma_j(\eta_x) \qquad \forall \eta\in {\mathbb{B}}_{n-1}(0,r)\,, \end{aligned}$$

and we claim that

$$\displaystyle \begin{aligned} C(x,R_j,r,\delta) \end{aligned}$$

is a coordinate cylinder for Ω around x and that γ _x represents ∂Ω in C(x, R _j, r, δ). To do so, we observe that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle { R_j(\varOmega-x)\cap ( {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[) } \\ &\displaystyle &\displaystyle =R_j\left( \varOmega-x^{(j)}-R_j^t(\eta_x,\gamma_j(\eta_x))^t \right)\cap ({\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[) \\ &\displaystyle &\displaystyle =\left( R_j(\varOmega-x^{(j)})-(\eta_x,\gamma_j(\eta_x))^t \right) \\ &\displaystyle &\displaystyle \qquad \qquad \cap \left( ( {\mathbb{B}}_{n-1}(\eta_x,r)\times]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x) +\delta[ )-(\eta_x,\gamma_j(\eta_x))^t \right) \\ &\displaystyle &\displaystyle =R_j(\varOmega-x^{(j)}) \cap \left( {\mathbb{B}}_{n-1}(\eta_x,r)\times]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x)+\delta[\right) -(\eta_x,\gamma_j(\eta_x))^t \\ &\displaystyle &\displaystyle =R_j(\varOmega-x^{(j)}) \cap \left( {\mathbb{B}}_{n-1}(0,r_j^{\prime})\times]-\delta_j^{\prime},\delta_j^{\prime}[ \right) \\ &\displaystyle &\displaystyle \qquad \qquad \cap \left( {\mathbb{B}}_{n-1}(\eta_x,r)\times]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x)+\delta[ \right)-(\eta_x,\gamma_j(\eta_x)) \\ &\displaystyle &\displaystyle =({\mathrm{hypograph}}_s(\gamma_j)) \\ &\displaystyle &\displaystyle \qquad \qquad \cap \left( {\mathbb{B}}_{n-1}(\eta_x,r)\times]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x)+\delta[ \right)-(\eta_x,\gamma_j(\eta_x))\,. \end{array} \end{aligned} $$

(10.23)

Next we observe that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle { {\mathrm{hypograph}}_s(\gamma_x) } \\ &\displaystyle &\displaystyle =\left\{ (\eta,y)\in {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[:\,y<\gamma_x(\eta) \right\} \\ &\displaystyle &\displaystyle =\left\{ (\eta,y)\in {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[:\, y+\gamma_j(\eta_x)<\gamma_j(\eta_x+\eta) \right\} \\ &\displaystyle &\displaystyle =\left\{ (\eta,y)\in {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[:\, (\eta+\eta_x,y+\gamma_j(\eta_x))\in {\mathrm{hypograph}}_s(\gamma_j) \right\} \\ &\displaystyle &\displaystyle =\left\{ (\eta,y)\in {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[:\, (\eta,y)\in {\mathrm{hypograph}}_s(\gamma_j)-(\eta_x,\gamma_j(\eta_x)) \right\} \\ &\displaystyle &\displaystyle =\bigg\{\bigg. (\eta,y)\in \left( {\mathbb{B}}_{n-1}(\eta_x,r)\times ]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x)+\delta,[ \right)-(\eta_x,\gamma_j(\eta_x)):\, \\ &\displaystyle &\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\eta,y)\in {\mathrm{hypograph}}_s(\gamma_j)-(\eta_x,\gamma_j(\eta_x)) \bigg.\bigg\} \\ &\displaystyle &\displaystyle =\bigg[\bigg.{\mathrm{hypograph}}_s(\gamma_j) \\ &\displaystyle &\displaystyle \qquad \qquad \qquad \cap \left( {\mathbb{B}}_{n-1}(\eta_x,r)\times ]-\delta+\gamma_j(\eta_x),\gamma_j(\eta_x)+\delta,[ \right)\bigg.\bigg]-(\eta_x,\gamma_j(\eta_x))\,. \end{array} \end{aligned} $$

(10.24)

Then by combining (10.23) and (10.24) we obtain

$$\displaystyle \begin{aligned} R_j(\varOmega-x)\cap ( {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[) ={\mathrm{hypograph}}_s(\gamma_x)\,. \end{aligned}$$

By the definition of γ _x and by inequality (10.22), we have

$$\displaystyle \begin{aligned} \gamma_x(0)=0\,,\qquad |\gamma_x(\eta)|<\delta/2\qquad \forall \eta\in {\mathbb{B}}_{n-1}(0,r)\,. \end{aligned}$$

Moreover, γ _x has the same regularity of γ _j and if α > 0, we have

$$\displaystyle \begin{aligned} |\gamma_x: \overline{{\mathbb{B}}_{n-1}(0,r)} |{}_{ \alpha }\leq \|\gamma_j\|{}_{ C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r_j^{\prime})}) } \leq\sup_{l=1,\dots,k}\|\gamma_l\|{}_{ C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r_l^{\prime})}) } <+\infty\,, \end{aligned}$$

and thus the proof is complete. □

Lemma 10.5

Let Ω be an open subset of ${\mathbb {R}}^{n}$ . Let K be a compact subset of Ω. Then there exists an open bounded subset Ω ₁ of Ω of class C ^∞ such that

$$\displaystyle \begin{aligned} K\subseteq \varOmega_1\subseteq \overline{\varOmega_1}\subseteq \varOmega\,. \end{aligned}$$

If we further assume that K is connected, then we can take Ω ₁ to be connected.

For a proof, we refer to [DaLaMu19, Ch. 2], which contains a proof due to G. De Marco.¹