UDC 517.946.9

MATHEMATICS

L. I. KAMYNIN

ON THE LYAPUNOV–GÜNTER THEOREMS FOR SPECIAL HEAT POTENTIALS

(Presented by Academician S. L. Sobolev on 24 V 1967)

In paper (¹) (see also (², ³)) a theory was constructed for the smoothness of heat potentials of a simple and a double layer with densities distributed on noncylindrical surfaces, entirely analogous to the classical Lyapunov–Günter theory (see (⁴)) for harmonic potentials. In (⁵) (see also (⁶)) the construction of a similar theory was begun for the special heat potentials (P) and (Q) (introduced by M. Pani in (⁷)), which play an important role in the theory of boundary-value problems with an oblique derivative for a parabolic equation of the second order. The present note, consisting of two sections, contains a complete and systematic investigation in Hölder smoothness spaces of the special heat potential of a simple layer (P[\varphi]) and the heat potential (Q[\varphi]), which is the derivative of the potential (P[\varphi]) in an oblique direction. In § 1 the improving properties of the direct values (\overline{Q}[\varphi]) on noncylindrical surfaces of various smoothness are studied, while in § 2 the smoothness of the heat potential (P[\varphi]) in the closed domain (\overline{D}_T), having (\Gamma) as its lateral boundary, is studied. The notation and definitions of the author’s papers (¹, ⁵) are used in the note.

Let (D_T) be a bounded domain of the ((n+1))-dimensional Euclidean space ((x,t) \equiv (x_1,x_2,\ldots,x_n;t)), situated between two hyperplanes (t=0) and (t=T>0) and having an (n)-dimensional noncylindrical surface (\Gamma) as its lateral boundary. Consider the heat potentials

[
P(\bar{x},t)\equiv P[\varphi]\equiv
\int_0^t d\tau \iint_{\Gamma_\tau} P(\bar{x},t;y,\tau)\varphi(y,\tau)\,d\sigma_y(\tau),
\qquad
(\bar{x},t)\in D_T,
]

[
Q(\bar{x},t)\equiv Q[\varphi]\equiv \partial P(\bar{x},t)/\partial \nu(x,t),
\qquad
(x,t)\in \Gamma_t;
]

(\Gamma_\tau) is the section of the surface (\Gamma) by the hyperplane (t=\tau), on which a field of directions is given with unit vector
(\nu(x,t)={\nu_1(x,t),\ldots,\nu_n(x,t;0)}), lying in the section (\Omega_t\equiv D_T\cap{t=t}) and making an acute angle not exceeding (\pi/2-d_0) ((d_0>0)) with the normal (N(x,t)) interior with respect to (\Omega_t) at the point ((x,t)\in\Gamma_t). (P(\bar{x},t;y,\tau)) is the special fundamental solution, introduced by M. Pani in (⁷), of the (n)-dimensional heat-conduction equation corresponding to the field of oblique directions (\nu(x,t)).

§ 1. Improving properties of the direct values of the heat potential (\overline{Q}[\varphi]). The notation (\alpha'), (\alpha^0), (\alpha^*), (\beta') from (⁵) will be used.

Theorem 1. Suppose that for (\Gamma), (\nu), and (\varphi) the following conditions are satisfied: (\Gamma) is of type

[
\Pi_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha,\,\alpha/2},
\qquad
0<\alpha\le 1,
]

[
\nu_j\in H_{2m+1,\,\beta,\,\beta/2}^{m,\,1,\,(1+\beta)/2}(\Gamma),
\qquad
0<\alpha\le \beta\le 1,
]

[
\cos(\nu(x,t),N(x,t))\ge d>0,
\qquad
(x,t)\in\Gamma_t,\quad d=\mathrm{const},\quad 0\le t\le T,
\tag{1}
]

[
\varphi\in H_{2m-1,\,1,\,(1+\alpha)/2}^{m,\,\alpha,\,\alpha/2}(\Gamma),
]

where

[
\left|\partial^k\varphi(y,\tau)/\partial\tau^k\right|
\le
\left|\partial^m\varphi/\partial t^m\right|_{\alpha}\tau^{m-k+\alpha/2},
]
[
(y,\tau)\in\Gamma,\qquad k=0,1,2,\ldots,m.
]

Then for (m=1,2,\ldots) (for (m=0) see Lemma 4 (5)) for (\overline Q(x,t)) one has

[
\overline Q\in H_{2m+1,\alpha,\alpha^/2}^{m,1,(1+\alpha^)/2}(\Gamma),
]

where the Hölder constants have the form ((C)|\varphi|_{2m+\alpha}),

[
\left|\partial^k\overline Q(x,t)/\partial t^k\right|
\le
(C)|\varphi|_{2m+\alpha}t^{m-k+(1+\alpha)/2},
\qquad (x,t)\in\Gamma,\qquad k=0,1,\ldots,m,
]

and when

[
p=0,1,2,\ldots,m,\qquad l_j=0,1,\ldots,2(m-p),\qquad j=1,2,\ldots,k,
]
[
\sum_{j=1}^{k}l_j=2(m-p), \tag{2}
]

where (k=n-1),

[
\left|
\partial^{2m-p+l}\overline Q(x,t)/
\partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i^{l}
\right|
\le
(C)|\varphi|_{2m+\alpha}t^{(1+\alpha-l)/2},
\qquad l=0,1.
]

Theorem 2. Suppose that for (\Gamma), (\nu), and (\varphi) the following conditions are satisfied: (\Gamma) is of type

[
\mathcal L_{2m+3,\alpha,\alpha/2}^{m+1,1,(1+\alpha)/2},\qquad
0<\alpha\le 1\quad (m=0,1,2,\ldots),
]

[
\nu_j\in H_{2m+1,1,(1+\beta)/2}^{m+1,\beta,\beta/2}(\Gamma),
\qquad 0<\alpha\le\beta\le 1,\qquad j=1,2,\ldots,m
]

(with (1) satisfied),

[
\varphi\in H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\Gamma),
]

[
\left|\partial^{k+l}\varphi(y,\tau)/\partial\tau^k\partial y_i^l\right|
\le
\left|\partial^m\varphi/\partial t^m\right|_{1+\alpha}
\tau^{m-k+(1+\alpha-l)/2}
]

[
(y,\tau)\in\Gamma,\quad k=0,1,2,\ldots,m;\quad
i=1,2,\ldots,n-1;\quad l=0\ \text{for } k<m,
]
[
l=0,1\ \text{for } k=m.
]

Then for (m=0,1,2,\ldots) for (\overline Q(x,t)) one has

[
\overline Q\in H_{2m+1,1,(1+\alpha^)/2}^{m+1,\alpha^,\alpha^*/2}(\Gamma),
]

where the Hölder constants have the form ((C)|\varphi|_{2m+1+\alpha}),

[
\left|\partial^k\overline Q(x,t)/\partial t^k\right|
\le
(C)|\varphi|_{2m+1+\alpha}t^{m-k+\alpha/2},
\qquad k=0,1,\ldots,m,
]

and, when (2) is satisfied, where (k=n-1),

[
\left|
\partial^{2m+1-p}\overline Q(x,t)/
\partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i
\right|
\le
(C)|\varphi|_{2m+1+\alpha}t^{(1+\alpha)/2},
]

[
\left(
\left|
\partial^{2m+1-p}\overline Q(x,t)/
\partial t^{p+1}\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}
\right|,
\right.
]

[
\left.
\left|
\partial^{2m+2-p}\overline Q(x,t)/
\partial t^p\partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\partial x_i\partial x_j
\right|
\right)
\le
(C)|\varphi|_{2m+1+\alpha}t^{\alpha/2}.
]

§ 2. Smoothness of the special heat potential of a simple layer (P[\varphi]) in the closed domain (\overline D_T)

Theorem 3. Suppose that for (\Gamma), (\nu) the conditions of Theorem 1 are satisfied,

[
\varphi\in H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\Gamma)
\qquad \text{for } m=0,1,2,\ldots,
]

[
\partial^k\varphi(y,0)/\partial\tau^k\equiv 0,
\qquad k=0,1,\ldots,m,\qquad (y,0)\in\Gamma_0,
]

and, when (2) is satisfied, where (k=n-1),

[
\left(
\left|
\partial^{2m-p+l}\varphi(y,\tau)/
\partial\tau^p\partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}}\partial y_i^l
\right|
\le
(C)|\varphi|_{2m+1+\alpha}\tau^{(1+\alpha-l)/2},
\qquad l=0,1.
\right.
]

Then

[
P \in H_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha',\,\alpha'/2}(\overline{D}_T),
]

where the Hölder constants have the form ((C)|\varphi|_{2m+1+\alpha}), and (see (2), where (k=n))

[
\left|
\partial^{2m+1+l-p}P(\bar{x},t)/\partial t^p \partial \bar{x}1^{l_1}\cdots
\partial \bar{x}_n^{l_n}\partial \bar{x}_i
\right|
\leq
(C)|\varphi|,} t^{(1+\alpha-l)/2
\quad l=0,1,
]

[
\left|
\partial^{2m+1-p}P(\bar{x},t)/\partial t^{p+1}\partial \bar{x}1^{l_1}\cdots
\partial \bar{x}_n^{l_n}
\right|
\leq
(C)|\varphi|,} t^{\alpha/2
\quad
(\bar{x},t)\in \overline{D}_T
]

(for (m=0), Theorem 3 coincides with Lemma 6 of (⁵)).

Theorem 4. Suppose that for (\Gamma) and (\nu) the conditions of Theorem 2 are fulfilled, and

[
\varphi \in H_{2m+1,\,1,\,(1+\alpha)/2}^{m+1,\,\alpha,\,\alpha/2}(\Gamma)
\quad \text{for } m=0,1,2,\ldots,
]

where

[
\partial^k \varphi(y,0)/\partial \tau^k = 0,
\quad
k=0,1,2,\ldots,m+1,
]

and (see (2), where (k=n-1))

[
\left|
\partial^{2m+1+l-p}\varphi(y,\tau)/\partial \tau^p
\partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}}
\partial y_i \partial y_j
\right|
\leq
(C)|\varphi|_{2m+2+\alpha}\tau^{(1+\alpha-l)/2},
\quad l=0,1;
]

[
\left|
\partial^{2m+1-p}\varphi(y,\tau)/\partial \tau^{p+1}
\partial y_1^{l_1}\cdots \partial y_{n-1}^{l_{n-1}}
\right|
\leq
(C)|\varphi|_{2m+2+\alpha}\tau^{\alpha/2}.
]

Then

[
P \in H_{2m+3,\,\alpha',\,\alpha'/2}^{m+1,\,1,\,(1+\alpha)/2}(\overline{D}_T),
]

where the Hölder constants have the form ((C)|\varphi|_{2m+2+\alpha}), and, when (2) is fulfilled, where (k=n) and (m) is replaced by (m+1),

[
\left|
\partial^{2(m+1)+l-p}P(\bar{x},t)/\partial t^p
\partial \bar{x}1^{l_1}\cdots \partial \bar{x}_n^{l_n}\partial \bar{x}_i
\right|
\leq
(C)|\varphi|,}t^{(1+\alpha-l)/2
\quad l=0,1.
]

The proofs of Theorems 1–4 are carried out by the methods of the papers (², ³, ⁶).

Received
22 V 1967

REFERENCES

¹ L. I. Kamynin, DAN, 160, No. 2, 271 (1965).
² L. I. Kamynin, Differential Equations, 1, No. 6, 799 (1965).
³ L. I. Kamynin, Differential Equations, 2, No. 5, 647 (1966).
⁴ N. M. Günter, The Theory of the Potential and Its Application to the Basic Problems of Mathematical Physics, Moscow, 1953.
⁵ L. I. Kamynin, DAN, 169, No. 4, 761 (1966).
⁶ L. I. Kamynin, Differential Equations, 2, No. 10, 1333 (1966); 2, No. 11, 1484 (1966).
⁷ M. Pagni, Ann. Scuola norm. sup. Pisa, Ser. III, II, fasc. I–II, 73 (1957).