Abstract
In this article, an elliptic equation, which type degenerates (either weakly or strongly) at the axis of a 3-dimensional cylinder, is considered. The statement of a Dirichlet type problem in the class of smooth functions is given and, subject to the type of degeneracy, the exact classical solutions are obtained. The uniqueness of the solutions is proved.
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This paper is a continuation of [1] and generalizes Problem D1 for equation (2) which is solved in this article. Namely, we consider here the more general Dirichlet problem in cylinder Q with non-zero boundary value conditions on the lateral surface of cylinder Q (including its edges).
In this part, we use the notation introduced in [1] and continue the numbering of assertions, remarks and formulas.
1 Statement of the problem and preliminaries
In this paper, we deal with the following Dirichlet problem.
Problem D2
Find the solution u of equation (2),
in the class of the functions \(C^{2} ( Q_{0} ) \cap C(\overline{Q}\backslash\{r=0\})\) (or, maybe, in the class \(C^{2} ( Q_{0} ) \cap C( \overline{Q})\)) which is bounded in \(Q_{0}\) and satisfies the boundary value conditions
and
for \((r,\varphi)\in D_{0}\cup K\) (or, maybe, for \((r,\varphi)\in \overline{D}\)); here \(f\in C (\overline{S} ) \) and \(f_{i}\in C ( \overline{D} ) \) are given functions such that the compatibility condition
are fulfilled.
Besides, we assume here that f and \(f_{i}\) are 2π-periodic functions in φ. (The concrete requirements concerning the smoothness of these functions will be given below.)
Let us note that the difference between Problems D1 and D2 is only this: in Problem D2 the compatibility condition (48) is more general than in Problem D1, where this condition is of the shape
As is mentioned in [1], the partial case of problem (46), (47) when \(f_{i}(r,\varphi)\equiv0\) and \(f ( r, ( i-1 ) H ) =0 \), \(i=1,2\), is investigated in [2]. Here, assuming that \(f\in C^{2}(\overline{S})\), the exact solution \(u_{0}\in C^{2} ( Q _{0} ) \cap C(\overline{Q})\) is composed; specifically, it is of the shape
here \(\alpha_{mn}^{0}\), \(\beta_{mn}^{0}\) are the coefficients of the Fourier expansion
where \(\gamma_{n}=\frac{\pi n}{H}\), \(R_{mn}\) are bounded at the point \(r=0\) solutions of the equation
i.e.,
where \(I_{\nu}\) and \(K_{\nu}\) are the modified Bessel functions of the first and second kind of order ν [3]. It follows directly from (49) that
and
for each \((\varphi,z)\in\overline{S}\), i.e., the solution \(u_{0}\) is continuous on the line of degeneracy \(r=0\).
2 Problem D2 with zero valued boundary conditions on the edges of cylinder Q
Primarily, we consider Problem D2 under the particular condition
instead of the condition (48). So, we seek the solution of this problem which is equal to zero on the two edges of cylinder Q. To this end, we shall use the results obtained in [1, 2].
Theorem 4
Let the functions f and \(f_{i}\), \(i=1,2\), be such that:
-
(i)
\(f\in C^{2}(\overline{S})\);
-
(ii)
\(f_{i}\) and \(\frac{\partial f_{i}}{\partial \varphi}\in C(\overline{D})\), \(\frac{\partial f_{i}}{\partial r} \in C ( D_{0}\cup K ) \), and
$$ \int _{0}^{R}\biggl\vert \frac{\partial f_{i}(r,\varphi)}{ \partial r}\biggr\vert \,\mathrm{d}r< \infty\quad \forall\varphi\in{}[-\pi,\pi], i=1,2; $$ -
(iii)
\(f_{i}(0,\varphi)=0\) \(\forall\varphi\in [ - \pi,\pi ] \), \(\frac{\partial^{2}f_{i}}{\partial r^{2}}\in C ( D_{0}\cup K )\), and
$$ \frac{\partial f_{i}(r,\varphi)}{\partial r}=O \bigl( r^{1-2\alpha} \bigr)\quad\textit{and}\quad \frac{\partial^{2}f_{i}(r,\varphi)}{\partial r^{2}}=O \bigl( r^{-2\alpha} \bigr)\quad\textit{as } r\rightarrow0, $$uniformly with respect to φ (in the case when \(\alpha<1\));
-
(iv)
\(\int_{0}^{R}r^{\frac{1-3\alpha}{2}}\,\mathrm{d}r \int_{-\pi}^{\pi} \vert f_{i}(r,\varphi)\vert \, \mathrm{d}\varphi<\infty\) (in the case when \(\alpha\geq1\));
-
(v)
the compatibility condition (52) holds.
Then there exists the unique solution u of Problem D2 such that:
-
(a)
\(u\in C^{2} ( Q_{0} ) \cap C(\overline{Q}\backslash \{r=0\})\) for \(\alpha<1\) under conditions (i)-(ii), (v), and for \(\alpha=1\) under conditions (i), (ii), (iv), (v);
-
(b)
\(u\in C^{2} ( Q_{0} ) \cap C(\overline{Q})\) for \(\alpha<1\) under conditions (i)-(iii), (v), and for \(\alpha>1\) under conditions (i), (ii), (iv), (v).
Proof
Let \(u_{0}\) be the solution of equation (2) defined by (49). Then, as is mentioned above,
and \(u_{0}\in C^{2} ( Q_{0} ) \cap C(\overline{Q})\). Denote by \(u^{(0)}\) the solution of Problem D1. (Remember that, subject to the type of degeneracy of equation (2), \(u^{(0)}\) is obtained under assumptions (ii)-(iv) in [1] (see Theorems 1-3).) Then the sum
satisfies equation (2) in \(Q_{0}\) and, since
this sum satisfies boundary value conditions (46), (47), evidently. Further, by virtue of (52), the equalities
hold, i.e., the sum \(u=u_{0}+u^{(0)}\) is equal to zero on the edges of cylinder Q.
Further, as is shown in [1], we have the inclusion \(u^{(0)}\in C^{2} ( Q_{0} ) \cap C(\overline{Q}\backslash\{r=0\})\) for \(\alpha=1\) (under conditions (ii), (iv)) and also for \(\alpha<1\) (under conditions (ii)), and the inclusion \(u^{(0)}\in C^{2} ( Q _{0} ) \cap C(\overline{Q})\) for \(\alpha<1\) (under conditions (ii), (iii)) and also for \(\alpha>1\) (under conditions (ii), (iv)). Hence, there hold the same inclusions for the solution u of Problem D2 as for the solution \(u^{(0)}\) of Problem D1, i.e., both assertions (a) and (b) of the lemma are true.
The uniqueness of Problem D2 follows under the condition (52) by virtue of Lemma 4 [1]. □
3 Problem D2 with non-zero conditions on the ages of cylinder Q
If the condition (52) is not fulfilled, then a solution of Problem D2 cannot be obtained in the same way as above. In this case we consider the auxiliary problem.
Problem D A
Find the solution \(u_{a}\in C^{2} ( Q_{0} ) \cap C(\overline{Q})\) of equation (2) which satisfies the boundary value conditions
where the function \(f\in C^{2} ( \overline{S} ) \) is such that \(f ( \varphi, ( i-1 ) H )\neq0\), \(i=1,2\), identically.
Assuming that the assumption (i) is fulfilled we expand the function f by the double Fourier series
where
According to the assumption \(f\in C^{2} ( \overline{S} ) \), this series converges uniformly and absolutely in S̅ [4]. By the method of separated variables, we represent the solution \(u_{a}\) of Problem D A by the series
where \(R_{mn}(r)\), \(m\in\mathbb{N} _{0}\), are the solutions of equation (50) defined by (51) for all \(n\geq0\) if \(c\neq0\), and for all \(n\geq1\) if \(c=0\), whereas
if \(c=0\). Hence, in the case when \(c=0\), the series (55) can be rewritten as follows:
where
It is easily seen that \(u_{a}(R,\varphi,z)=f(\varphi,z)\). Since the functions \(R_{mn}(r)\), \(m,n\in\mathbb{N} _{0}\), are monotonically decreasing on the interval \((0,R)\) for all \(\alpha>0\), i.e.,
by virtue of the absolute convergence of the series (54), we have the estimate
Thus, the series (55) converges uniformly and absolutely in Q̅; consequently, \(u_{a}\in C(\overline{Q})\).
The inclusion \(u_{a}\in C^{2}(Q_{0})\) can be justified just in the same way as, for instance, the same inclusion of the sum \(u_{1}(r,\varphi,z)\) of the series (34) in [1]. Thus, \(u_{a}\in C^{2}(Q_{0})\cap C( \overline{Q})\). Besides, let us note that the component \(u_{a}^{(1)}\) of sum (56) is harmonic in D and continuous in D̅.
Therefore, the following theorem holds.
Theorem 5
Let \(f\in C^{2}(\overline{S})\). Then there exists a solution \(u_{a}\in C^{2}(Q_{0})\cap C( \overline{Q})\) of Problem D A , which can be represented by the series (55).
Next, we consider the behavior of the derivatives of solution \(u_{a}\) as \(r\rightarrow0\).
Lemma 5
Let the inclusion \(f\in C^{2}(\overline{S})\) holds. Then the solution (55) of Problem D A is such that
for all \(\alpha>0\).
Proof
Under the assumption of the lemma, equality (55) is term-by-term differentiable with respect to φ. Particularly,
besides, due to the assumption \(f\in C^{2}(\overline{S})\), the series on the right-hand side converges also uniformly and absolutely. Thus,
Since the functions \(R_{mn}(r)\) are monotonically decreasing on the interval \(( 0,R ) \) for all \(\alpha>0\), we obtain
i.e., the derivative \(\frac{\partial u_{a}}{\partial\varphi }\) is continuous in D̅ for all \(\alpha>0\). □
Lemma 6
Let \(\alpha<1\). Assume that the inclusion \(f\in C^{2}(\overline{S})\) holds. Then the solution (55) of Problem D A is such that
uniformly in S̅.
Proof
We deal with the asymptotic properties of the derivative \(R_{mn}^{ \prime}\) as \(r\rightarrow0\). Since \(\alpha<1\), (see (51)) for all \(n\geq0\) if \(c\neq0\), and for all \(n\geq1\) if \(c=0\),
and
if \(c=0\). Taking into account that [3, 5]
and
by straightforward calculation, we obtain for the following asymptotic expressions as \(r\rightarrow0\):
If \(c=0\), then, obviously,
Thus, if \(\frac{1}{2}<\alpha<1\), then we have the relation
for all \(n\in\mathbb{N} _{0}\).
Differentiating equality (55) with respect to r and taking into account (58), we obtain
Observe that \(I_{0} ( 2\mu_{n}R^{1-\alpha} ) \rightarrow \infty\) as \(n\rightarrow\infty\) (because of the asymptotic properties of the function \(I_{0} ( z ) \) as \(z\rightarrow+\infty\) [5]). Therefore, the series on the right-hand side of the last equality converge uniformly and absolutely, i.e.,
uniformly in S̅. Since \(\alpha>1\), this implies (57).
If \(0<\alpha\leq\frac{1}{2}\), then the asymptotic relations (58) do not hold. However, in this case
for all \(n\in\mathbb{N} _{0}\). Therefore, it follows from (55) that
uniformly in S̅. Since \(\alpha<1\), this yields (57). □
Lemma 7
Let \(\alpha\geq1\). Assume that inclusion \(f\in C^{2}(\overline{S})\) holds. Then the solution (55) of Problem D A satisfies the condition (57) and, if \(c\neq0\), it satisfies the condition
If \(c=0\), then the condition (59) is satisfied if and only if
with \(m_{0}= [ \frac{3(\alpha-1)}{2} ]\), where \([a]\) is the integer part of a number \(a\in\mathbb{R} \).
Proof
Let \(\alpha>1\). Then (see (51))
if \(n^{2}+c^{2}\neq0\). Since [3, 5]
and
we have the expressions
\(m,n\in\mathbb{N} _{0}\), where \(\nu_{n}\) and \(\nu_{n}^{\prime}\) are some non-zero constants which depend only on n. Observe that
as \(r\rightarrow0\). Thus, if \(c\neq0\), then we get from (55)
where
Hence, we have the estimates
uniformly in S̅. This yields the obvious validity of the conditions (57), (59).
If \(c=0\), then the solution \(u_{a}\) can be represented as the sum (56). Taking into account that
by just the same reasoning as above, we get for the component \(u_{a}^{(2)}\) the estimates
uniformly in S̅. Thereby, the component \(u_{a}^{(2)}\) also satisfies the two conditions (57), (59). The component \(u_{a}^{(1)}\), as harmonic in the desk D function, satisfies the condition (59), obviously, whereas, as is easily seen, it satisfies (59) if and only if \(\alpha_{m0}=\beta_{m0}=0\) for \(m\leq m_{0}= [ \frac{3(\alpha-1)}{2} ] \), where, let us remember, \(\alpha_{m0}\) and \(\beta_{m0}\) are the Fourier coefficients of expansion (54):
Thus, in order for the condition (59) to be fulfilled, orthogonality conditions (60) are sufficient and necessary.
Let \(\alpha=1\). Then
and, consequently,
uniformly for \((\varphi,z)\in\overline{D}\). Thus, the condition (57) is fulfilled if \(c\geq0\), whereas the condition (59) is valid only for \(c>0\).
If \(c=0\), then
Hence, the condition (59) is fulfilled if and only if \(\alpha_{00}=0\), i.e., if (60) with \(m_{0}=0\) holds. □
Let us couple the assertions of Lemmas 5-7 by the following corollary.
Corollary
If \(f\in C^{2}(\overline{S})\), then the functions
are such that \(h_{i}\) and \(\frac{\partial h_{i}}{\partial\varphi} \in C(\overline{D})\), \(\frac{\partial h_{i}}{\partial r}\in C ( D _{0}\cup K ) \), and
for each \(\alpha>0\). Furthermore, if \(\alpha\geq1\) and \(c\neq0\), then
If \(\alpha\geq1\) and \(c=0\), then this estimate holds if and only if function f satisfies (60). Thus, if \(f\in C^{2}(\overline{S})\) and the condition (60) for \(\alpha\geq1\) and \(c=0\) is fulfilled, then the functions \(h_{i}\), \(i=1,2\), satisfy the same conditions (ii) and (iv) of Theorem 4 as the functions \(f_{i}\), \(i=1,2\).
Theorem 6
Let the following assumptions hold:
-
1.
Functions f and \(f_{i}\), \(i=1,2\), satisfy the corresponding conditions (i) and (ii) of Theorem 4, and the compatibility condition (48) is fulfilled.
-
2.
In the case when \(\alpha\geq1\), the functions \(f_{i}\), \(i=1,2\), satisfy the condition (iv) of Theorem 4.
-
3.
In the case when \(\alpha\geq1\) and \(c=0\) function f satisfies the condition (60).
Then, under assumption 1, Problem D2 has the solution \(u\in C^{2}(Q_{0})\cap C ( \overline{Q}\backslash\{r=0\} )\) for \(\alpha<1\). If \(\alpha\geq1\) and \(c\neq0\), then, under both assumptions 1 and 2, Problem D2 has the solution u from the class \(C^{2}(Q _{0})\cap C ( \overline{Q}\backslash\{r=0\} ) \) for \(\alpha=1\), and from the class \(C^{2}(Q_{0})\cap C ( \overline{Q} ) \) for \(\alpha>1\). If \(\alpha\geq1\) and \(c=0\), then the last predication is true under assumptions 1-3. In all cases the solution is unique.
Proof
Introduce the functions
and consider Problem D1 to equation (2) (see (3)-(4) in [1]) with the following boundary value conditions:
Let us verify whether the functions \(g_{i}\), \(i=1,2\), satisfy the respective conditions of Theorems 1-3 (see [1]).
By virtue of Theorem 5, under conditions (i), there exists the solution \(u_{a}\in C^{2}(Q_{0})\cap C(\overline{Q})\) of Problem D A . This yields the correct determination of the functions \(h_{i}\) by (61). According to the corollary, the condition (i) implies the guarantee that the functions \(h_{i}\) satisfy the same conditions (ii), (iv) of Theorem 4 as the functions \(f_{i}\), \(i=1,2\). Consequently, the functions \(g_{i}\), \(i=1,2\), defined by (62), also satisfy the conditions (ii), (iv). Moreover,
because of (48). Therefore, these functions (61) satisfy the respective conditions of Theorems 1-3 [1], according to those there exists the unique solution \(u^{(0)}\) of problem (63), (64), which is from the class:
-
(i)
\(C^{2}(Q_{0})\cap C ( \overline{Q}\backslash\{r=0\} ) \) if \(\alpha<1\) (under assumption 1 of the theorem);
-
(ii)
\(C^{2}(Q_{0})\cap C ( \overline{Q}\backslash\{r=0\} ) \) if \(\alpha=1\) (under assumptions 1 and 2 in the case when \(c\neq0\) and under assumptions 1-3 in the case when \(c=0\));
-
(iii)
\(C^{2}(Q_{0})\cap C ( \overline{Q} ) \) if \(\alpha>1\) (under assumptions 1 and 2 in the case when \(c\neq0\) and under assumptions 1-3 in the case when \(c=0\)).
Obviously,
is the solution of equation (2) which belongs to the same classes (i)-(iii) as the solution \(u^{(0)}\) and is such that
for \(( r,\varphi ) \in D_{0}\cup K\) if \(\alpha\leq1\) and for \(( r,\varphi ) \in\overline{D}\) if \(\alpha>1\), and
i.e., boundary value conditions (46), (47) are fulfilled. Thus, (65) represents the solution of Problem D2.
The uniqueness of the solution follows from the maximum principle,
which due to Lemma 4 holds. □
It follows from Theorems 1-3 [1] that, under conditions of Theorem 5, the solution \(u^{(0)}\) of problem (63), (64) can be expanded by the corresponding series (34), (41), and (45) subject to the type of degeneracy of equation (2). Since solution \(u_{a}\) of Problem D A is given analytically by (55), the solution (65) of Problem D2 can be represented exactly.
Remark 1
If \(\alpha<1\), then it follows from (55) that
where
Hence, \(g_{i}(0,\varphi)=0\), \(i=1,2\), only in the peculiar occurrence; specifically, if and only if \(f_{i}(0,\varphi)=f_{0i}=\mathrm{const}\) and if
In the opposite case, \(g_{i}(0,\varphi)\neq0\), \(i=1,2\), i.e., the functions \(g_{i}\), \(i=1,2\), do not satisfy all conditions of Theorem 1 under which there exists the solution of Problem D1 (63), (64) continuous at the line of degeneracy \(r=0\). Therefore, if \(\alpha< 1\) and the condition (52) is not satisfied, the solution of Problem D2 does not belong to \(C^{2}(Q_{0}) \cap C ( \overline{Q} ) \) in general.
4 Conclusions concerning Problem D2
It is easily seen that Problem D2 generalizes Problem D1. Nevertheless, the main difficulties in the consideration of Dirichlet type problem assert in Problem D1, where the behavior of boundary functions at the points \(P_{0}\) and \(P_{H}\) must be harmonized with the type of the degeneracy of equation (2). However, the following effects in the case of Problem D2 are evident (see Theorems 4 and 6):
-
1.
If the boundary value conditions are non-zero valued on the edges of cylinder Q, then the solution of this problem is continuous on the line of degeneracy \(r=0\) only if \(\alpha>1\), i.e., only in the case of the strong degeneracy of equation (2).
-
2.
If boundary value conditions are zero valued on the edges of cylinder Q, then, under supplementary requirements for boundary functions on the bases of this cylinder (see conditions (iii) of Theorem 4), the solution of Problem D2 can be also continuous on the line \(r=0\) if \(\alpha<1\), i.e., if equation (2) is weakly degenerate.
-
3.
If \(c=0\) and \(\alpha\geq1\), then the solvability of Problem D2 requires some additional orthogonality condition on the lateral surface of cylinder Q (see (60) in Lemma 7) with respect to the boundary function.
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Rutkauskas, S. Exact solutions of Dirichlet type problem to elliptic equation, which type degenerates at the axis of cylinder. II. Bound Value Probl 2016, 182 (2016). https://doi.org/10.1186/s13661-016-0691-7
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DOI: https://doi.org/10.1186/s13661-016-0691-7