The Weighted Square Integral Inequalities for Smooth and Weak Subsolution of Fourth Order Laplace Equation

1. Introduction

Among the most important of all partial differential equations are undoubtedly second order Laplace equation. Some of practical problems also covered by higher order Laplace equation. In [1], [2], [3], [4] and [5] the author develop the energy estimates for convex functions, 4-convex functions and also for super-harmonic functions. These estimates are very important in financial mathematics one can see [6], so it is also interesting to develop similar results for subsolution of fourth order Laplace equation.

The fourth order Laplace equation with \(n\) variables is given as Let us denote $$\Delta^4 \equiv \frac{\partial^4}{\partial x^4_1}\,+\,\frac{\partial^4}{\partial x^4_2}\,+\ldots+\,\frac{\partial^4}{\partial x^4_n}$$ Then (1) becomes as The function \({u(x)}\in C^4(D)\) is called subsolution (suppersolution) of fourth order Laplace equation (2) if The bounded measurable function \({u(x)}\) is called week subsolution of (1) if \({u(x)}\) satisfy It is trivial that \(\Delta^4\) is self-adjoint operator i.e. $$\Delta^4=\Delta^{*4}$$ Through out the paper we will use the following notations. $$grad\,\, {u(x)} = \big(\frac{\partial {u(x)}}{\partial x_1},\frac{\partial {u(x)}}{\partial x_2},\ldots,\frac{\partial {u(x)}}{\partial x_n}\big)$$ $$grad^2\,\, {u(x)} = \big(\frac{\partial^2{{u(x)}}}{\partial x^2_1},\frac{\partial^2{{u(x)}}}{\partial x^2_2},\ldots,\frac{\partial^2{{u(x)}}}{\partial x^2_n}\big)$$ We will organize the paper in the following way in second section we will derive the energy estimate for the fourth order Laplace equation. Also we approximate the week subsolution by the smooth ones. In the last section we will derived similar estimate regarding week subsolution.

2. The Weighted Energy Estimates for the Smooth Subsolution for the Fourth Order Laplace Equation.

Lemma 2.1. [5] Assume \(f(x)\in C^4(I)\) is the smooth and \(h(x)\) is the smooth non-negative weight function having compact support. From the proof of Theorem 2.1 in [5] one can derive the identity in Lemma 2.1 as:

Theorem 2.2. Let \({u}_i(x)\in C^4(D),\, i=\,1,2\) be the smooth subsolutions of (2) over the domain \( D\subseteq\mathbb{R}^n\) and also \(\frac{\partial^2 u_i}{\partial x_j^2}\geq 0\,\,\,\,\forall\,\,\,j=1, 2,\ldots, n\), then we have the following energy estimate for the difference of the functions

where \(h(x)\) is the non-negative smooth weight function satisfying

Proof. Let

Take $$\int\limits_D\left|grad^2{u(x)}\right|^2\,\,h(x)\,\,dx=\int\limits_D\bigg[\big(\frac{\partial^2{u(x)}}{\partial x^2_1}\big)^2+\,\big(\frac{\partial^2 {u(x)}}{\partial x^2_2}\big)^2\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\ldots+\,\big(\frac{\partial^2 {u(x)}}{\partial x^2_n}\big)^2 \bigg]\,h(x)\,dx$$ \[=\int\limits_D \big(\frac{\partial^2 {u(x)}}{\partial x^2_1}\big)^2\,h(x)\,dx+\,\int\limits_D\big(\frac{\partial^2 {u(x)}}{\partial x^2_2}\big) ^2\,h(x)\,dx+\ldots+\] By Lemma 2.1, the above may write as $$=\sum^n_{j=1}\big(\int\limits_D{u(x)}\frac{\partial^4{u(x)}}{\partial x^4_j}\,h(x)dx-2\int\limits_D{u(x)}\frac{\partial^2{u(x)}}{\partial x^2_j}\frac{\partial^2 h(x)}{\partial x^2_j}\,dx+\frac{1}{2}\int\limits_D\, u^2(x)\,\frac{\partial^4h(x)}{\partial x^4_j}dx\big)$$ $$\leq \int\limits_D\left|{u(x)}\right|\left|\sum^n_{j=1}\frac{\partial^4{u(x)}}{\partial x^4_j}\right|h(x)dx+\,2\,\int\limits_D|{u(x)}|\left|\sum^n_{j=1}\frac{\partial^2{u(x)}}{\partial x^2_j}\,\frac{\partial^2 h(x)}{\partial x^2_j}\right|\,dx$$ $$+\,\frac{1}{2}\,\int\limits_D\, u^2(x)\,\sum^n_{j=1}\frac{\partial^4h(x)}{\partial x^4_j}\,dx$$ $$\leq\,\,\sup_{x\in D}\left|{u(x)}\right|\,\int\limits_D\,\left|\sum^n_{j=1}\frac{\partial^4{u(x)}}{\partial x^4_j}\right|\,h(x)\,dx\,+\,2\,\sup_{x\in D}|{u(x)}|\,\int\limits_D\,\left|\sum^n_{j=1}\,\frac{\partial^2{u(x)}}{\partial x^2_j}\,\frac{\partial^2 h(x)}{\partial x^2_j}\right|\,dx$$ $$+\,\frac{1}{2}\,\int\limits_D\, u^2(x)\,\sum^n_{j=1}\frac{\partial^4h(x)}{\partial x^4_j}\,dx\,$$ Now replacing \({u(x)}={u}_2(x)-{u}_1(x)\) we obtain, $$\int\limits_D\left|grad^2\big({u}_2(x)-{u}_1(x)\big)\right|^2\,\,h(x)\,\,dx\,$$ $$\leq\,\,\sup_{x\in D}\left|\big({u}_2(x)-{u}_1(x)\big)\right|\,\int\limits_D\,\left|\sum^n_{j=1}\frac{\partial^4}{\partial x^4_j}\big({u}_2(x)-{u}_1(x)\big)\right|\,h(x)\,dx\,$$ $$+\,2\,\sup_{x\in D}|\big({u}_2(x)-{u}_1(x)\big)|\,\int\limits_D\,\left|\sum^n_{j=1}\,\frac{\partial^2}{\partial x^2_j}\big({u}_2(x)-{u}_1(x)\big)\,\frac{\partial^2 h(x)}{\partial x^2_j}\right|\,dx$$ $$+\,\frac{1}{2}\,\int\limits_D\, \big({u}_2(x)-{u}_1(x)\big)^2\,\sum^n_{j=1}\frac{\partial^4h(x)}{\partial x^4_j}\,dx\,$$ Again using the definition of weight function and integration by parts formula, we obtain $$\int\limits_D\,\sum^n_{j=1}\frac{\partial^4 {u}_i(x)}{\partial x^4_j}\,h(x)\,dx\,=\,\int\limits_D\ {u}_i(x)\,\sum^n_{j=1}\,\frac{\partial^4h(x)}{\partial x^4_j}\,dx\,\,\,\,\,\,\,\,\,i=1,2$$ and $$\int\limits_D\sum^n_{j=1} \frac{\partial^2{u}_i(x)}{\partial x^2_j} \, \frac{\partial^2h(x)}{\partial x^2_j}\,dx\,=\,\int\limits_D\ {u}_i(x)\,\sum^n_{j=1}\,\frac{\partial^4h(x)}{\partial x^4_j}\,dx\,\,\,\,i=1,2$$ the above (10) becomes $$\int\limits_D\left|grad^2\big({u}_2(x)-{u}_1(x)\big)\right|^2 h(x) dx\leq$$ $$\int\limits_D \big(\frac{1}{2}{\big({u}_2(x)-{u}_1(x)\big)^2}- \sup_{x\in D}\left|{u}_2(x)-{u}_1(x)\right|\big({u}_2(x)+{u}_1(x)\big)\big) \sum^n_{j=1}\frac{\partial^4h(x)}{\partial x^4_j}\,dx$$

Remark 2.1. Taking the supremum norm on above inequality we obtained

Remark 2.2. The above estimates over domain \(D \in \mathbb{R}^n\) also holds for arbitrary ball \(B(x_o,r)\) with center \(x_o\) and radius \(r\).

From now we use ball \(B(x_o,r)\) as domain in \(\mathbb{R}^n\).

3. The Weighted Energy Estimates for the Week Subsolution of Fourth Order Laplace Equation.

The continuous function \({u(x)}\) is said to be week subsolution of (2) if where $$\psi\in C^\infty_c(B).$$ Now we will approximate the week subsolution of (2) by the smooth ones. For this we will use the classical mollification. Define where \(x\in\mathbb{R}^n\) and \(c>0\) is such that Now we define Let us denote By the definition \(\eta_{\epsilon}(x-y)\), It is trivial that so, where \(\Delta^4_x\) and \( \Delta^4_y\) are the fourth order Laplace operator w.r.t \(x\) and \(y\) respectively.
Let the ball \(B_k=B(x_o,r_k)\) with \(r_k=\frac{k+1}{k+2}\,r\,\,\,\,\,,k=1,2,.\) The next theorem tells about the existence of smooth approximation.

Theorem 3.1. Let \({u(x)}\) be the continuous week subsolution of (2) and convex over the ball \(B.\) Then for any \(k=1,2,3,\ldots,\,\, \text{there exists}\,\,\,\,\hat{\epsilon}>0\) such that \( 0< \epsilon< \hat{\epsilon}\) each function \({u}_\epsilon(x)\) is smooth convex over the ball \(B_k\) and also $$\Delta^4 {u}_\epsilon(x)\geq 0\,\,\,\,\text{if}\,\,\,\,x\in B_k.$$

Proof. For fixed \(k=1,2,\ldots,\)
Let

By the definition, it is trivial that \({u}_\epsilon(x)\) is infinitely differentiable and also from [7] \({u}_\epsilon(x)\) is smooth convex for each of its arguments.
we next claim that \(\eta_\epsilon(x-y)\) has compact support in the ball \(B\).
Take another ball \(\hat{B}_k\) in the following way $$If\,\,\,\,y\notin \hat{B}_k\,\,\,\,\,then$$ Hence \(\eta_\epsilon(x-y)\) has compact support.
Hence by the definition of week subsolution and also using (22), we get We will defined \(h_k(x)\) as: $$ \begin{cases} h_k(x)>0 & if\,\,\,\, x\in B_k(x_0, r_k)\\ h_k(x)=0 & if\,\,\,\, x\in \partial B_k(x_0, r_k) \end{cases} $$ where \(r_k=\frac{k+1}{k+2}r\)

Theorem 3.2. Let \({u(x)}\) be the continuous week subsolution of (2) and convex over the ball \(B\) then it possesses the following weak partial derivatives \(\frac{\partial^2{{u(x)}}}{\partial x^2_i} ,\,\,\, i=1, 2,\ldots,n \) over the ball \(B\).

Proof. For the existence of first derivative \(\frac{\partial {u(x)}}{\partial x_i} \,\,\,i=1, 2, . . . ,n \) one can see [4]. Let us suppose the mollification \({u}_\epsilon(x)\) defined in (15) for the week subsolution of fourth order Laplace equation \({u(x)}\).
For the continuous function \({u(x)}\), the ball \(B,\) it is well-known fact that on compact set \(\subseteq B \) we have the following uniform-convergence $$\sup\limits_k\left|{u}_\epsilon(x)-{u(x)}\right|\underset{\epsilon\to0}{\longrightarrow}0.$$ Let us denote \({u}_\epsilon(x)\) by \({u}_m(x)\) for \(\epsilon=\frac{1}{m},\,\,m=1, 2, . . . \) so above becomes

The ball \(B_k\) are compactly contained in the original ball \(B\)\\ From the theorem 3.1, we know that for any k=1, 2, ...., \(\text{there exists}\,\,\, m_k\) such that \({u}_m(x)\) is smooth subsolution of (2)
Take the ball \(B_{k+l}\) and write the inequality (11) for $${u}_1(x)={u}_m(x)\,\,\,and\,\,\,{u}_2(x)=u_p(x)$$ Let us denote $$\alpha_{k+l}=\int\limits_{B_{k+l}} \left|\Delta^4 h_{k+l}\right| dx,\,\,\,$$ $$\hat{\alpha}=\inf\limits_{x\in B_{k+l}}\,\,h_{k+l},$$ then (25) becomes where \(\hat{\alpha}\neq 0\). Writing the left hand integral for the smaller ball \(B_k\), we have From (24), we have $$\left\|{u}_p-{u}_m\right\|_{L^\infty_{(B_{k+l})}}\longrightarrow0\,\,\,\,\,\text{as}\,\,\,\,\,\,\,\,m,p\longrightarrow\infty$$ so (27) becomes. $$\lim_{m,p\rightarrow\infty} \,\,\,\sum^n_{i=1}\int\limits_{B_k}\big(\frac{\partial^2{u}_p}{\partial x^2_i}-\frac{\partial^2{u}_m}{\partial x^2_i}\big)dx=0 $$ The completeness of \(L^2(B_k) \) ensure the convergence of above sequence. So there exist a class of measurable functions \(v_{k,i}(x)\in L^2(B_k)\) such that $$\sum^n_{i=1} \int\limits_{B_k}\big(\frac{\partial^2{{u}}_m}{\partial x^2_i}-v_{k,i}\big)^2 dx\underset{m\to\infty}{\longrightarrow}0 , k=1, 2,...$$ we extend \(v_{k,i}(x)\) trivially outside the ball \(B_k\) by \(0\).
Let us denote $$v_i(x)=\lim_{k\rightarrow\infty}\, \sup_{x\in D} \,\,\,v_{k,i}, \,\,\,i=1,2,...,n$$ It can be checked easily that $$v_i(x)=v_{k,i}$$ for the ball \(B_k.\)
Next we claim that \(v_i(x)\) represent the weak second order partial derivative \(\frac{\partial^2 {u(x)}}{\partial x^2_i}\) of \({u(x)}\).
Take \(\psi(x)\) an arbitrary function having compact support in B.
Then suppose \(\psi(x)\subset B_k\) from some k.
We have $$\int\limits_B \frac{\partial^2{u}_m}{\partial x^2_i} \,\,\psi dx\,\,=\,\,\int\limits_B {u}_m \,\,\,\frac{\partial^2\psi}{\partial x^2_i} \,\,\,dx $$ for the integers \(m\geq m_k\).\\ But we have the following convergence $$\left| {u}_m(x)-{u(x)}\right|\underset{m\to\infty}{\longrightarrow}0 \,\,\text{on} \,\,\,B_k$$ and $$ \left\| \frac{\partial^2{{u}}_m}{\partial x^2_i}-v_i(x)\right\|_{L^2(B_k)}\underset{m\to\infty}{\longrightarrow}0$$ using these Limits, the above becomes. $$\int\limits_{B_k} v_i(x)\,\,\psi(x)\,\,dx = \int\limits_{B_k} {u(x)} \frac{\partial^2\psi}{\partial x^2_i}\,\,dx.$$ This shows that \(v_i(x)\,\,\, i=1,2,...,n\) are the weak partial derivative of \({u(x)}\). Rewriting the inequality (11) for the functions \({u}_1(x)=0\) and \({u}_2(x)={u}_m(x)\) for \(m\geq m_{k+l}\) over the ball \(B_{k+l}\), we get $$\int\limits_{B_{k+l}}\left|grad^2 {u}_m(x)\right|^2 h_{k+l}(x)dx\leq \frac{3}{2}\alpha_{k+l}\left\|{u}_m(x)\right\|^2_{L^\infty(B_{k+l})}$$ Taking limit \(m\rightarrow \infty,\) the above becomes $$\int\limits_{B_{k+l}}\left|grad^2 {u}_m(x)\right|^2 h_{k+l}(x)dx\leq \frac{3}{2}\alpha_{k+l}\left\|{u(x)}\right\|^2_{L^\infty(B_{k+l})}$$ Now restricting the left hand side for the smaller ball \(B_{k},\) we have $$\int\limits_{B_k}\left|grad^2 {u}_m(x)\right|^2 h_{k+l}(x)dx\leq \frac{3}{2}\alpha_{k+l}\left\|{u(x)}\right\|^2_{L^\infty(B_{k+l})}$$ Now making limit as \(m \rightarrow \infty,\) we obtain $$\int\limits_{B_k}\left|grad^2 {u(x)}\right|^2 h(x)dx\leq \frac{3}{2}\alpha_{\infty}\left\|{u(x)}\right\|^2_{L^\infty(B)}< \infty.$$ The left hand side of above increases as \(k\) increases and also bounded, so it will have finite limit, i.e which completes the proof.

Now we prove the inequality for the weak subsolution of fourth order Laplace equation.

Theorem 3.3. Let \({u_i(x)}, i=1,2\) be the continuous weak subsolution of (2), and also it satisfies $$\frac{\partial^2{{u(x)}}}{\partial x^2_j}\geq 0\,\,\,\,\,\,\,\,\,\forall j=1,2,\ldots,n$$ then the following is valid

where \(h(x)\) is the weight function satisfying (7).

Proof. Take mollification \({u}_{m,i}(x)\), for \(i=1,2 \) of the continuous weak subsolution \({u}_i(x)\), for \(i=1,2 \) respectively.
Since for the ball \(B_{k+l}\), \(\text{there exists}\) integer \(m_{k+l},\) such that each function \({u}_{m,i}\), for \(i=1,2\) is the smooth subsolution in the ball \(B_{k+l},\) if \(m>m_{k+l}\). Also we have the following convergence $$\left\|{u}_{m,i}(x)-{u}_i(x)\right\|_{L^{\infty}(B_{k+l})}\underset{m\to\infty}{\longrightarrow}0 \,\,\,\, i= 1,2.$$ Now writing the inequalities the function \({u}_{m,i}(x)\) , i=1,2 and the ball \(B_{k+l}\) , we get

Taking Limit as \(m\rightarrow\infty\) , the latter inequality (30) becomes Again writing the above inequality, the left hand side for smaller ball \(B_k\) and also taking Limit \(l\rightarrow\infty\), we get, we finally obtain The above inequalities holds for all \(B_k\), k=1,2,3,...,n. So also true for ball B.
Which completes the proof.

Conflict of Interests

The authors hereby declare that there is no conflict of interests regarding the publication of this paper.