Abstract Generated abstract
The paper studies differential properties of solutions to a Dirichlet problem in the half-plane for higher-order equations with even derivatives in two variables. Using Fourier separation of variables, it derives integral representations of solutions for cases including s = 1, 2, and 3, with generalized Poisson kernels, and estimates these kernels through Hardy, Hölder, and Minkowski inequalities. The main results give sufficient, sharp regularity conditions on boundary functions in Sobolev-type H and W classes that imply corresponding anisotropic regularity of the solution, as well as weighted summability of higher derivatives. The paper also applies these results to obtain conditions ensuring boundedness of a generalized Dirichlet integral, explicitly formulated for s = 2.
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MATHEMATICS
Ya. S. BUGROV
DIFFERENTIAL PROPERTIES OF SOLUTIONS OF CERTAIN HIGHER-ORDER DIFFERENTIAL EQUATIONS
(Presented by Academician L. I. Sedov, 5 VII 1962)
In this work we investigate the differential properties of solutions of certain boundary-value problems depending on the differential properties of the boundary functions in terms of the \(H\)-classes and \(W\)-classes of S. L. Sobolev and generalized classes.
We consider the Dirichlet problem for the equation
\[ L^{(l,s)}u \equiv (-1)^{l+1}\frac{\partial^{2l}u}{\partial x^{2l}} +(-1)^{s+1}\frac{\partial^{2s}u}{\partial y^{2s}}=0 \tag{1} \]
(\(l,s\) are natural numbers, \(y>0\), \(-\infty<x<\infty\)) under the conditions
\[ \left.\frac{\partial^j u}{\partial y^j}\right|_{y=0} =\varphi_j(x)\qquad (j=1,2,\ldots,s-1) \tag{2} \]
and the function \(u(x,y)\) is bounded in \(R_2^0=\{y>0;\ -\infty<x<\infty\}\).
The conditions (2) are understood in the sense that
\[ \lim_{y\to +0}\int_{-\infty}^{\infty} \left| \frac{\partial^j u(x,y)}{\partial y^j}-\varphi_j(x) \right|^p dx=0 \qquad (1<p<\infty). \tag{3} \]
Let us note that for \(l=s=1\), \(L^{(1,1)}u\equiv \Delta u\), where \(\Delta\) is the Laplace operator. Applying the Fourier method of separation of variables, we obtained an analytic representation of the solution of the boundary-value problem (1)—(2). In particular, for \(s=1\) the solution of the boundary-value problem has the form
\[ u(x,y)=\int_{-\infty}^{\infty}K_0(t,y;l,1)\varphi_0(x+t)\,dt, \tag{4} \]
where
\[ K_0(t,y;l,1)=\frac{1}{2\pi}\int_0^\infty e^{-\lambda^l y}\cos\lambda t\,d\lambda; \]
for \(s=2\)
\[ u(x,y)=\int_{-\infty}^{\infty}K_0(t,y;l,2)\varphi_0(x+t)\,dt +\int_{-\infty}^{\infty}K_1(t,y;l,2)\varphi_1(x+t)\,dt, \tag{5} \]
where
\[ K_0(t,y;l,2)=\frac{1}{2\pi}\int_0^\infty e^{-\nu y}(\cos\nu y+\sin\nu y)\cos\lambda t\,d\lambda, \]
\[ K_1(t,y;l,2)=\frac{1}{2\pi}\int_0^\infty e^{-\nu y}\frac{\sin\nu y}{\nu}\cos\lambda t\,d\lambda \qquad \left(\nu=\frac{\sqrt{2}}{2}\lambda^{l/2}\right); \]
for \(s=3\)
\[ u(x,y)=\int_{-\infty}^{\infty} K_0(t,y;l,3)\varphi_0(x+t)\,dt+ \int_{-\infty}^{\infty} K_1(t,y;l,3)\varphi_1(x+t)\,dt+ \]
\[ +\int_{-\infty}^{\infty} K_2(t,y;l,3)\varphi_2(x+t)\,dt, \]
where
\[ K_0(t,y;l,3)=\frac{1}{2\pi}\int_0^\infty \left[ e^{-y\lambda^\alpha}+\frac{2\sqrt{3}}{3}e^{-\frac{y}{2}\lambda^\alpha} \sin\frac{\sqrt{3}}{2}y\lambda^\alpha \right]\cos\lambda t\,d\lambda \quad \left(\alpha=\frac{l}{3}\right), \]
\[ K_1(t,y;l,3)= \]
\[ =\frac{1}{2\pi}\int_{-\infty}^{\infty}\lambda^{-\alpha} \left[ e^{-y\lambda^\alpha}+e^{-\frac{y}{2}\lambda^\alpha} \left( \sqrt{3}\sin\frac{\sqrt{3}}{2}y\lambda^\alpha -\cos\frac{\sqrt{3}}{2}y\lambda^\alpha \right) \right]\cos\lambda t\,d\lambda, \]
\[ K_2(t,y;l,3)= \]
\[ =\frac{1}{2\pi}\int_{-\infty}^{\infty}\lambda^{-2\lambda} \left[ e^{-y\lambda^\alpha}+e^{-\frac{y}{2}\lambda^\alpha} \left( \frac{\sqrt{3}}{3}\sin\frac{\sqrt{3}}{2}y\lambda^\alpha -\cos\frac{\sqrt{3}}{2}y\lambda^\alpha \right) \right]\cos\lambda t\,d\lambda. \]
Using the explicit representation of the solution of the boundary-value problem, on the basis of the properties of the kernels \(K_j(t,y;,l,s)\) \((j=0,1,\ldots,s-1)\), which we call generalized Poisson kernels, and with the aid of the Hardy, Hölder, and Minkowski inequalities, we proved a number of theorems generalizing the corresponding results of S. M. Nikol’skii \((^1)\), O. V. Besov \((^3)\), S. V. Uspenskii \((^{5,6})\), L. D. Kudryavtsev \((^7)\), and others.
Let \(R_1=\{-\infty<x<\infty\}\), \(R_2(0,1)=\{0<y<1,\ -\infty<x<\infty\}\).
Theorem 1. If the functions \(\varphi_j(x)\in H_p^{\left[r+(s-1-j)\frac{l}{s}\right]}(R_1)\) \((r>0;\ 1<p<\infty;\ j=0,1,\ldots,s-1)\), then the solution of the boundary-value problem (1)—(2)
\[ u(x,y)\in H_{px}^{\left[r+\left(s-1+\frac{1}{p}\right)\frac{l}{s}\right]}\,[R_2(0,1)] \]
for any \(r>0\);
\[ u(x,y)\in H_{py}^{\left(\frac{sr}{l}+s-1+\frac{1}{p}\right)}[R_2(0,1)] \]
for \(p=2\), \(r\) arbitrary; for \(p\ne2\), \(r+(s-1-j)\frac{l}{s}\ne 2k\) \((2k\) an even number\()\)*.
Theorem 2. If the functions \(\varphi_j(x)\in W_p^{\left[r+(s-1-j)\frac{l}{s}\right]}(R_1)\) \((1<p<\infty;\ j=0,1,\ldots,s-1;\ r+(s-1-j)\frac{l}{s}\) are not integers for \(p\ne2\)), then the solution of problem (1)—(2)
\[ u(x,y)\in W_p^{(r_1,r_2)}[R_2(0,1)], \]
where
\[ r_1=r+\left(s-1+\frac{1}{p}\right)\frac{l}{s},\qquad r_2=\frac{s}{l}r_1 \]
**.
\[ \underline{\phantom{xxxxxxxxxxxxxxxx}} \]
* For the definition of the classes \(H_p^{(r_1,r_2)}\), see \((^2)\).
** For the definition of the classes \(W_p^{(r_1,r_2)}\), see \((^{8,4,5})\).
The proof of Theorems 1 and 2 was carried out by us for \(s=1,2,3\) and arbitrary natural \(l\). These theorems cannot be improved in these terms, as follows from the results of S. M. Nikol’skii \(({}^{2})\), p. 296, and O. V. Besov \(({}^{4})\), p. 79.
Further, we have shown that the solution of problem (1)—(2) has derivatives of arbitrary order, summable with a weight. For example, for \(s=1\) the following holds.
Theorem 3. Let \(\varphi_0(x)\in W_p^{(r)}(R_1)\) \((r>0;\ r=\rho+d,\ 0<\alpha<1;\ \rho\) an integer\()\); \(\frac{r}{l}+\frac{1}{p}=n+\beta,\ 0\leqslant \beta<1;\ r+\frac{l}{p}=nl+l\beta=nl+k+\gamma,\ k=0,1,\ldots,l-1,\ 0\leqslant \gamma<1,\ n\) an integer.
Then the function (4) has derivatives of order higher than \(n\) with respect to \(y\) and of order higher than \(nl+k\) with respect to \(x\), summable with a weight:
\[ \left\| \frac{\partial^{\,n+s_1+s_2}}{\partial y^{\,n+s_1}\partial x^{s_2}}\, y^{\,s_1-\beta+\frac{s_2}{l}} \right\|_{L_p[R_2(0,1)]} \leqslant c\|\varphi_0\|_{W_p^{(r)}(R_1)}, \]
where \(s_1=1,2,\ldots;\ s_2=0,1,\ldots;\)
\[ \left\| \frac{\partial^{\,nl+k+s_1+s_2}u}{\partial y^{s_1}\partial x^{\,nl+k+s_2}}\, y^{\,s_1+\frac{s_2-\gamma}{l}} \right\|_{L_p[R_2(0,1)]} \leqslant c\|\varphi_0\|_{W_p^{(r)}(R_1)}, \]
where \(s=0,1,\ldots;\ s_2=1,2,\ldots,\)
\[ \|\psi\|_{L_p[R_2(0,1)]} = \left( \int_0^1\int_{-\infty}^{\infty}|\psi|^p\,dx\,dy \right)^{\frac1p}. \]
This result is definitive and generalizes the result of S. V. Uspenskii \(({}^{6})\) (Theorem 2).
Using Theorems 1 and 2, one can obtain sufficient conditions for the boundedness of the generalized Dirichlet integral
\[ D(u;l,s)_p = \int_0^\infty\int_{-\infty}^{\infty} \left[ \left|\frac{\partial^l u}{\partial x^l}\right|^p + \left|\frac{\partial^s u}{\partial y^s}\right|^p \right]\,dx\,dy \]
in terms of \(H\)-classes and \(W\)-classes. We give these conditions for \(s=2\).
Theorem 4. Let the functions \(\varphi_0\in W_n^{\left(\frac{l}{2q}+\frac{l}{2}\right)}(R_1)\), \(\varphi_1\in W_n^{\left(\frac{l}{2q}\right)}(R_1)\), where \(\frac1p+\frac1q=1\) (for \(p\ne2\), \(\frac{l}{2q},\ \frac{l}{2q}+\frac{l}{2}\) are not integers).
Then these functions can be continued to \(R_2^0\) in the form of the solution (5), with \(D(u;l,2)_p<\infty\), and conditions (2) are satisfied.
In terms of \(H\)-classes, a similar theorem is formulated with accuracy up to an arbitrary \(\varepsilon>0\).
Blagoveshchensk State
Pedagogical Institute
Received
28 VI 1962
References
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